T-030: $s (18) \geq 4679 / 1000 = 4.679$

V3 C3 lower bound confirmed superseded by T-102

2026-09-19 established · Levy after Burns, Massaccesi · $n = 18$

$s (18) \geq 4679 / 1000$ = 4.679, from this project's weighted fractional unavoidable-set certificate at container side 4679/1000. The case previously held 1871/400 = 4.6775 from T-029; the movement is +0.0015. The selected external report 9141/2000 = 4.5705 (GitHub; not replayed here) lies 0.1085 below this result.

One size, and not by monotonicity. Only Condition 2 mentions n among the five conditions, so an atom set of mass 71573611/4000000 = 17.89340275 certifies the side for every integer strictly above that mass, which is 18 and upward. It does not reach $n = 17$ , where T-019 still holds 459/100. From $n = 19$ on the register already holds 24/5 = 4.80, so the certificate is true there and weaker.

One rung is retained rather than a ladder. The side was reached by seeding the generator's site set with T-029's own atoms scaled from 1871/400 and adding a windows-5 lattice. Auto at 4679/1000 resolved to (32, 43, 54), the same triple T-029 used. $n = 18$ is off the H-218 sweep, so this retain confirms H-221, not H-218.

Significance, composition and next rung

Significance: Scored at S3 by the calibration note T-020 wrote for exactly this case: "Further sizes from the same generator, absent a new technique or a case that changes character, belong at S3." This is the same generator at $n = 18$ , one size-specific covering rather than T-019's Condition 2 carry, and $n = 18$ does not change character at 4.679. The movement is +0.0015 against T-029's verified field and 0.1085 against the strongest named public candidate. Held below S4 because the technique is the one T-017 already banked, the displacement is of this project's own prior rung rather than a published closed form, and $n = 18$ is not a central open case.
Composition: Primary at $n = 18$ : one certificate, one accepted verdict, the bound being the container side directly, with no monotonicity or composition step between the artifact and the claim. Two evidence entries whose method values differ decide it, the exact event-cell sweep and the interval branch and bound, both run in this repository on the frozen bytes; that is C3, with the two methods shown beside the rung. As with T-029 there is no independent evaluator and no review record, so rung 4 waits on two adversarial AI reviews and a human oversight record.
Next rung: The retained certificate's total is 71573611/4000000 = 17.89340275, leaving 426389/4000000 = 0.10659725 below eighteen, and that margin is what a further rung has to be found inside. The side above it, 117/25 = 4.68, locked at 18.000000 on the T-019-seeded construction, on T-027's own atoms, on T-028's own atoms, and on three earlier site sets; adding sites can still lower a restricted optimum, so 117/25 is not barred. The remaining interval to that plateau is 0.001. The method's ceiling is 5B = 4.9885, and the best known packing is 4.82287566, so the real runway is 0.1439 to the packing. V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained.

Superseded as the verified lower bound on 2026-10-02 by wand125's replayed $s (18) \geq 939 / 200$ (T-045), so a further rung of this method moves the case only above 4.695; stays true as stated.
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
$s (18) \geq 4426213 / 1000000$ , 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 this result
$s (18) \geq 4679 / 1000 = 4.679$
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-030: s(18)≥4679/1000=4.679s(18)≥4679/1000=4.679

The case

Results on the case

Links

T-030: $s (18) \geq 4679 / 1000 = 4.679$