T-019: $s (n) \geq 459 / 100 = 4.59$ for $n = 17, 18, 19$

V3 C3 lower bound confirmed superseded by T-093, T-102 and T-103

2026-09-04 established · Levy after Burns, Massaccesi · $n = 17, 18, 19$

$s (17) \geq 459 / 100$ , and $s (18) \geq 459 / 100$ and $s (19) \geq 459 / 100$ , from this project's weighted fractional unavoidable-set certificate at container side 459/100 = 4.59.

This displaces Massaccesi's 22529/5000 = 4.5058 (T-015) and the two cases T-016 carried from it, all three of which had held that value until 2026-09-04. The movement past the published value is +0.0842 at each of the three sizes.

A stronger public candidate, anabologyco-maker's side 4.5705 (nine thousand one hundred forty-one two-thousandths) of 16 August 2026 (GitHub; not replayed here), was outside the record's search corpus when this was registered and was found on 2026-09-07; it lies 0.0195 below this result. Two rungs are retained below it, 451/100 and 229/50.

The September 2026 DS7 audit identifies a stronger reported n19 bound from Theorem 10 at k=4, approximately 4.6172815, with an unresolved theorem/table discrepancy and missing proof. T-019 does not improve that report; its public source displacement is supported at n17 and n18, while all three certified inequalities and their movements from the prior register remain valid.

The three sizes do not need a monotonicity step. Only Condition 2 mentions n among the five conditions, so an atom set of mass 423327/25000 certifies the side for every integer above it, which is 17 and upward; monotonicity would give the same thing and is not what is used.

It does not improve $n = 20$ , where Nagamochi's 1 + sqrt(13) = 4.6055... is larger -- the certificate does prove $s (20) \geq 459 / 100$ , but that is the weaker statement.

Significance, composition and next rung

Significance: Scored against the rubric's anchor for S4, "a reusable technique, bound family, or resolved disputed value". One certificate moves three registered cases, and it replaces a published value rather than an inherited or closed-form one -- the only bound this project has displaced that someone else had put in print.

A stronger public candidate found afterwards, anabologyco-maker's 9141/2000 = 4.5705 of 16 August 2026 (GitHub; not replayed here), was outside the search corpus at scoring time, so the displacement at n17 and n18 is 0.0195 over that public report and 0.0842 over the one this record had adopted. At n19, the DS7 audit now retains a stronger source-reported bound than T-019; its 0.0842 movement refers to the prior register. The family-level score is unchanged.

Held below S5 because none of $n = 17$ , 18, 19 is a central open case, the movement is 0.0842 rather than a qualitative change, weighted resource counting predates this project, the recent pure-atomic rational direction-net architecture follows Burns, the LP parameter line follows Massaccesi, and the case does not change character at the new value.
Composition: Primary at all three sizes: one certificate, one accepted verdict, the bound being the container side directly. Its mass 423327/25000 is below seventeen, and Condition 2 is the only condition that mentions n. The same certificate therefore applies directly at n17, n18 and n19; monotonicity would also prove the latter two inequalities but is unnecessary for this certificate.
Next rung: A second machine method decides it, the interval-certified decision, the first here that is not exact-algebraic. V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained.

The bound itself: the retained certificate's total is 423327/25000 = 16.933080, leaving 0.066920 below seventeen, and that margin is what a further rung has to be found inside.

One figure from the ladder is worth carrying, because it contradicts the obvious reading. Margin is not monotone in the side. The 451/100 rung two below this one has total 16.593620 and margin 0.406380 -- an order of magnitude more headroom at a side 0.08 lower -- and the 229/50 rung between them has margin 0.034265, half of what 459/100 has at a higher side. So the covering value at a given side is set by the site set at least as much as by the side itself, and a better site set at a higher side can open the margin back up rather than close it. The $n = 12$ ladder shows the same shape in the same direction: margin 0.007175 at 197/50, then 0.029410 at the higher 79/20.

Massaccesi's own atoms are the reference point and not a limit: they give a covering value of 203/12 at 4.5058, tight by construction, which bounds nothing about what a different site set reaches at a larger side.

A further rung is therefore a search question, and the quantity to read is the restricted optimum a converged run reaches, never an extrapolated rate.

One such question was asked and answered on 2026-09-04, at $n = 18$ and side 117/25 = 4.68, and it is recorded here as measurement rather than as a claim about tau*. Three site sets in succession -- 538, 578 and 618 orbits -- returned a restricted optimum of exactly 18.000000, the third of them after 157 row-generation rounds that grew the row set from 15888 to 27516 and cost 7056 s. Adding sites can only lower a restricted optimum, and across those three it did not move at all.

Two readings are open and the run did not separate them. Either the covering value at that side is at or above eighteen, in which case no certificate exists there; or the site sets are still short of it and the optimum is sitting on a degenerate vertex, which the collapse of pricing from 90 s to 1--3 s suggests -- a dual that sparse is what a degenerate basis looks like. Exactly round values are the known artefact signature in this pipeline, 18.0 having been one for $n = 17$ 's grid-31 optima, so the second reading cannot be dismissed.

What is not in doubt is the cost of settling it: two hours for the round that did not move. The run was stopped for that reason rather than for its answer, and the side is available to a later block with a cheaper decision path.

The method's own ceiling is not close here. A certificate for n cannot exist above ceil(sqrt(n)) * B, a wider container holding ceil(sqrt(n))^2 pairwise disjoint axis-parallel B-squares whose masses Condition 5 forces past n; for $n = 17$ that is 5B = 4.9885, so 459/100 still has 0.3985 of runway. Unlike $n = 12$ , where the ceiling sits below the conjectured value and forecloses the case, nothing structural stops this one.

Where the next rung is worth spending on, though, is not $n = 17$ . Of the five conditions only Condition 2 mentions n, and the covering program behind the search does not contain n at all -- minimising total mass subject to every admissible B-square carrying mass at least 1 is a question about L, B and the net. So one atom set proves $s (n)$ >= L for every integer n above its mass, this certificate's 16.933080 reaching 17, 18 and 19 directly, and a larger n is strictly easier at the same side.

The headroom said the rest, when this was written. $n = 17$ had 0.0855 to its best known packing at 4.675530; $n = 18$ had 0.2329 to 4.822876 and $n = 19$ had 0.2956 to 4.885618, both still carrying this certificate's side in the verified register; and $n = 20$ had 0.3944 to the trivial 5, holding Nagamochi's closed-form 1 + sqrt(13) = 4.605551. A run at a side whose covering value lands between 17 and 18 would raise $n = 18$ and leave $n = 17$ where it is; one landing between 19 and 20 would displace a published closed form.

The resumed search ran the same day and is registered as T-020, at side 24/5 with retained certificate mass 18.922620, between eighteen and nineteen. It supersedes this result at $n = 19$ and moves $n = 20$ and $n = 21$ as well. What this result still holds alone is $n = 17$ and $n = 18$ : T-020's atoms are too heavy for either, which is the reach rule above read in the other direction. This result stays true as stated at all three of its sizes and keeps its rungs.

For $n = 17$ the reading is unchanged and the runway is 0.3985 to the ceiling and 0.0855 to the packing. For $n = 18$ , T-030 now certifies 4679/1000; T-029's 1871/400 remains as the previous rung. 117/25 remains the open probe this next_rung named.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps

The cases

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…

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…

Case record

4.824.886

456

4.3595.359

\sqrt{n}

\sqrt{n} + 1

Proven

$4.82500$ $\leq s (19) \leq$ $4.885619$

exact

Citation record n-019

lowerwand125 after Tokoharu, Levy et al. 2026, GitHub (confirmed T-103)

upperWainwright 1979, Squares in Squares

Open

optimality

The case record

LowerUpper

Verified

\frac{193}{40}

3 + \frac{4}{3} \sqrt{2}

Reported

\frac{193}{40}

3 + \frac{4}{3} \sqrt{2}

Gap

\frac{4 \sqrt{2}}{3} - \frac{73}{40}

≈ 0.06061808…

Results on these cases

33 results in the register on $n = 17, 18, 19$ , oldest first, each with what it established and how it stands now.

2005 published T-007 · $n = 17, 18, 19$
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
V0 C1 lower bound incomplete on these cases, superseded by T-093, T-102 and T-103
Nagamochi · Nagamochi 2005 · source · register
2026-08-21 published T-015 · $n = 17$
$s (17) \geq 22529 / 5000 = 4.5058$
V3 C3 lower bound confirmed superseded by T-093
Massaccesi after Burns · Burns–Massaccesi n17 · packet · register
2026-08-21 published T-016 · $n = 18, 19$
$s (n) \geq 22529 / 5000$ for $n = 18, 19$ , by monotonicity from T-015
V3 C3 lower bound confirmed superseded by T-102 and T-103
Massaccesi after Burns · Burns–Massaccesi n17 · packet · register
2026-08-31 established T-001 · $n = 17$
$s (17) \geq 4426213 / 1000000 = 4.426213$ , from a sixteen-point unavoidable set
V3 C3 lower bound confirmed superseded by T-093
Levy after Bentz · register
2026-08-31 established T-002 · $n = 18$
$s (18) \geq 4426213 / 1000000$ , by monotonicity from T-001
V3 C3 lower bound confirmed superseded by T-102
Levy after Bentz · register
2026-08-31 established T-003 · $n = 17, 18$
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 this result · $n = 17, 18, 19$
$s (n) \geq 459 / 100 = 4.59$ for $n = 17, 18, 19$
V3 C3 lower bound confirmed superseded by T-093, T-102 and T-103
Levy after Burns, Massaccesi · register
2026-09-04 established T-020 · $n = 19$
$s (n) \geq 24 / 5 = 4.80$ for $n = 19, 20, 21$
V3 C3 lower bound confirmed superseded by T-103
Levy after Burns, Massaccesi · register
2026-09-04 published T-085 · $n = 17, 18, 19$
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 · $n = 18$
$s (18) \geq 467 / 100 = 4.67$
V3 C3 lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-19 established T-028 · $n = 18$
$s (18) \geq 187 / 40 = 4.675$
V3 C3 lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-19 established T-029 · $n = 18$
$s (18) \geq 1871 / 400 = 4.6775$
V3 C3 lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-19 established T-030 · $n = 18$
$s (18) \geq 4679 / 1000 = 4.679$
V3 C3 lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-20 published T-032 · $n = 17$
$s (17) \geq 461300 / 99999 = 4.61304613 \dots$ , 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 · $n = 17$
$s (17) > 461300 / 99853 = 4.6197910929 \dots$
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 · $n = 17$
$s (17) \leq 4.6755300936045509 \dots$ , 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 · $n = 17$
$s (17) > 231001 / 50000 = 4.62002$
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 · $n = 17$
$s (17) > 232001 / 50000 = 4.64002$
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 · $n = 17$
$s (17) > 466001 / 100000 = 4.66001$
V3 C3 lower bound confirmed superseded by T-093
Kleddamag after Levy, Mira, Guzhou0806 · Kleddamag n17 4.66001 · packet · source · review · register
2026-09-27 published T-045 · $n = 18, 19$
Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$
V3 C3 lower bound confirmed superseded by T-102 and T-103
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · packet · packet · source · review · register
2026-09-27 published T-046 · $n = 18, 19$
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
V0 C0 lower bound recorded superseded by T-102 and T-103
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · register
2026-09-28 published T-042 · $n = 17$
$s (17) > 233009 / 50000 = 4.66018$
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 · $n = 17$
$s (17) > 116511 / 25000 = 4.66044$
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 · $n = 17, 18, 19$
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 · $n = 17, 18, 19$
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
V3 C3 lower bound confirmed on these cases, superseded by T-093, T-102 and T-103
Karakuş · Karakuş 2026 · source · register
2026-09-30 published T-093 · $n = 17$
$s (17) > 18641771 / 4000000 = 4.66044275$
V3 C3 lower bound confirmed
Guzhou0806 after Kleddamag, Mira, Levy · Guzhou0806 n17 R071 · packet · source · review · register
2026-10-01 published T-068 · $n = 19$
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
V3 C3 lower bound confirmed on these cases, superseded by T-103
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register
2026-10-01 published T-074 · $n = 19$
Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$
V3 C3 lower bound confirmed on these cases, superseded by T-103
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · packet · source · review · register
2026-10-05 published T-096 · $n = 18$
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 · $n = 18$
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-100 · $n = 19$
Mixed rectangle-measure lower bound verified at $n = 19$ , past the best 18-square packing
V3 C3 lower bound confirmed superseded by T-103
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds finer net 2026-10-06 · packet · source · review · register
2026-10-06 published T-102 · $n = 18$
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
2026-10-06 published T-103 · $n = 19$
Mixed rectangle-measure lower bound verified at $n = 19$ , 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-019: s(n)≥459/100=4.59s(n)≥459/100=4.59 for n=17,18,19n=17,18,19

The cases

Results on these cases

Links

T-019: $s (n) \geq 459 / 100 = 4.59$ for $n = 17, 18, 19$