T-020: $s (n) \geq 24 / 5 = 4.80$ for $n = 19, 20, 21$

V3 C3 lower bound confirmed superseded by T-052, T-103 and T-104

2026-09-04 established · Levy after Burns, Massaccesi · $n = 19, 20, 21$

$s (19) \geq 24 / 5$ , $s (20) \geq 24 / 5$ and $s (21) \geq 24 / 5$ , from this project's weighted fractional unavoidable-set certificate at container side 24/5 = 4.80.

Two of the three cases previously held Nagamochi's 2005 general closed form in this register's independently verified lower fields -- min(ceil(sqrt(N)), sqrt(N - 2*floor(sqrt(N)) + 1) + 1), which gives 1 + sqrt(13) = 4.6055... at $n = 20$ and 1 + sqrt(14) = 4.7416... at $n = 21$ -- and the third held the 459/100 = 4.59 this project certified earlier the same day as T-019. The movement is +0.21 at $n = 19$ , +0.194449 at $n = 20$ and +0.058343 at $n = 21$ , relative to those verified fields.

The September 2026 DS7 audit records stronger external reports below 4.80 separately, with missing source proofs and table discrepancies explicit; no claim of historical priority follows from the earlier source omissions.

The three sizes take no monotonicity step. Only Condition 2 mentions n among the five conditions, so an atom set of mass 946131/50000 certifies the side for every integer strictly above that mass, which is 19 and upward. From $n = 22$ on the register already holds 5, so the certificate is true there and weaker; these three are where it moves anything.

One rung is retained rather than a ladder. The side was reached in a single resumed column-generation run, not climbed to, so there is no weaker certificate at this size to compare it against.

Significance, composition and next rung

Significance: Scored against the rubric's anchor for S4, "a reusable technique, bound family, or resolved disputed value", and by parity with T-019, which has the same shape: one certificate moving three registered cases and displacing a published value.

This one displaces more. The movement is +0.21 at $n = 19$ against T-019's +0.0842, the largest single-case movement in the register; and at $n = 20$ and $n = 21$ it displaces the peer-reviewed 2005 closed form previously used in the verified register. The September 2026 source audit corrects the earlier claim that no size-specific bounds had been reported; it does not change these measured displacements of the verified fields.

Held below S5 for the reasons T-019 was: none of the three is a central open case, the weighted-resource lineage predates this project, the pure-atomic rational direction-net architecture follows Burns, the LP parameter line follows Massaccesi, and no case changes character at the new value.

A calibration note for whoever scores the next one. This is the fourth result from one instrument in a day, and the S4 anchor is being carried here by the displacement rather than by the technique, which T-017 already banked. Further sizes from the same generator, absent a new technique or a case that changes character, belong at S3.
Composition: Primary at all three sizes: one certificate, one accepted verdict, the bound being the container side directly. There is no monotonicity or composition step anywhere between the artifact and the claim -- the three sizes come out of Condition 2, which is the only condition that mentions n. 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; that is C3, with the two methods shown beside the rung. Unlike T-017 and T-019 no independent evaluator reviewed the scoring, and no review record is retained, so rung 4 waits on two adversarial AI reviews and a human oversight record.
Next rung: This result's 24/5 rung has total 18.922620, leaving 0.077380 below nineteen, and that margin is where a further rung had to be found. It is a wide margin by this register's standards -- $n = 17$ 's top rung has 0.066920 and $n = 12$ 's has 0.001040 -- which says the side was not pushed to where the covering value stops it, only to where the run was stopped. That is the honest reading of how this one was obtained.

The column generation was halted at round 9 with a restricted optimum of 18.916941, because four more rounds would have cost about 3.75 h to buy margin nothing needed. The side above it was never attempted.

Where the ceiling sits differs across the three sizes and it matters. A certificate for n cannot exist above ceil(sqrt(n)) * B, which for all three is 5B = 4.9885, so 0.1885 of structural runway remains. But at $n = 19$ the best known packing is 4.885618, and no certificate can exceed a side a packing achieves, so the real runway there is 0.085618. At $n = 20$ and $n = 21$ the best known packing is the trivial 5, so the ceiling binds first: the method could in principle close either to within 0.0115 of the upper bound and can never reach it. That asymmetry is the targeting instruction.

A run above 4.885618 that succeeded would contradict the retained $n = 19$ packing and is a refutation to look for rather than a rung to expect; a run between 4.80 and 4.9885 that succeeds moves $n = 20$ and $n = 21$ and stops at $n = 19$ . The quantity to read is the restricted optimum a converged run reaches, never an extrapolated rate, and the cost was the exhaustive tier: the 24/5 rung's exact sweep took 5378 s at its own 2260-atom set, against 173 s for the interval route on the same bytes. The same evening the sweep was rewritten to decide in integers on the weights' common scale, in parallel over directions, and returns the same least covered mass in 38.7 s on the same box.

The margin was found and taken the following day: T-021 certifies 97/20 = 4.85 at $n = 20$ and $n = 21$ from a site set seeded with this certificate's own atoms, and supersedes this result at both sizes. What this result still holds alone is $n = 19$ , whose bound stays 24/5 because T-021's atoms are too heavy for it; the rung itself is retained at cases/n20_fractional_certificate/certificate-24-5.json and replays by name. On 2026-10-02 wand125's replayed rectangle certificate, $s (19) \geq 963 / 200$ (T-045), superseded it at $n = 19$ as well. This result stays true as stated at all three of its sizes.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps

The cases

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…

Case record

4.905

456

4.4725.472

\sqrt{n}

\sqrt{n} + 1

Proven

$4.90500$ $\leq s (20) \leq$ $5$

exact

Citation record n-020

lowerwand125 after Tokoharu, Levy et al. 2026, GitHub (confirmed T-104)

Open

optimality

The case record

LowerUpper

Verified

\frac{981}{200}

Reported

\frac{981}{200}

Gap

\frac{19}{200}

= 0.095

Case record

456

4.5835.583

\sqrt{n}

\sqrt{n} + 1

Proven

$s (21) =$ $5$

optimal
exact

Citation record n-021

lowerDaniel after Burns, Massaccesi 2026, GitHub (confirmed T-052)

The case record

LowerUpper

Verified55

Reported55

Gap0 solved: the verified bounds meet

Results on these cases

22 results in the register on $n = 19, 20, 21$ , oldest first, each with what it established and how it stands now.

2005 published T-007 · $n = 19, 20, 21$
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
V0 C1 lower bound incomplete on these cases, superseded by T-052, T-103 and T-104
Nagamochi · Nagamochi 2005 · source · register
2026-08-21 published T-016 · $n = 19$
$s (n) \geq 22529 / 5000$ for $n = 18, 19$ , by monotonicity from T-015
V3 C3 lower bound confirmed superseded by T-103
Massaccesi after Burns · Burns–Massaccesi n17 · packet · register
2026-09-04 established T-019 · $n = 19$
$s (n) \geq 459 / 100 = 4.59$ for $n = 17, 18, 19$
V3 C3 lower bound confirmed superseded by T-103
Levy after Burns, Massaccesi · register
2026-09-04 established T-020 this result · $n = 19, 20, 21$
$s (n) \geq 24 / 5 = 4.80$ for $n = 19, 20, 21$
V3 C3 lower bound confirmed superseded by T-052, T-103 and T-104
Levy after Burns, Massaccesi · register
2026-09-04 published T-085 · $n = 19, 20, 21$
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
2026-09-05 established T-021 · $n = 20, 21$
$s (n) \geq 97 / 20 = 4.85$ for $n = 20, 21$
V3 C3 lower bound confirmed superseded by T-052 and T-104
Levy after Burns, Massaccesi · register
2026-09-23 established T-034 · $n = 21$
$s (21) \geq 122 / 25 = 4.88$
V3 C3 lower bound confirmed superseded by T-052
Levy after Burns, Massaccesi · register
2026-09-23 published T-050 · $n = 21$
$s (21) \geq 5000 / 1001 = 4.995004995 \dots$
V3 C3 lower bound confirmed superseded by T-052
Daniel after Burns, Massaccesi · evand square-packing 2026 · packet · source · review · register
2026-09-27 published T-045 · $n = 19, 20$
Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$
V3 C3 lower bound confirmed superseded by T-103 and T-104
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · packet · packet · source · review · register
2026-09-27 published T-046 · $n = 19, 20$
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
V0 C0 lower bound recorded superseded by T-103 and T-104
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · register
2026-09-27 published T-052 · $n = 21$
$s (21) = 5$ , by a mixed cover of points and grid-line segments
V3 C3 optimality confirmed
Daniel after Burns, Massaccesi · evand square-packing 2026-09-28 · packet · source · review · register
2026-09-28 published T-055 · $n = 21$
$s (21) = 5$ by a point-only route
V3 C3 simplification confirmed
wand125 after Daniel, Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point and mixed bounds 2026-09-28 · packet · source · review · register
2026-09-29 published T-058 · $n = 19, 20, 21$
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 · $n = 19, 20, 21$
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
V3 C3 lower bound confirmed on these cases, superseded by T-052, T-103 and T-104
Karakuş · Karakuş 2026 · source · register
2026-10-01 published T-068 · $n = 19, 20$
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 and T-104
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register
2026-10-01 published T-074 · $n = 19, 20$
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 and T-104
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · packet · source · review · register
2026-10-02 published T-077 · $n = 20$
Rectangle-density lower bounds replayed at $n = 20$ , 42 and 70
V3 C3 lower bound confirmed superseded by T-104
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-02 · packet · packet · source 1 · source 2 · review 1 · review 2 · register
2026-10-03 published T-081 · $n = 21$
$s (k^{2} - 4) = k$ for every integer $k$ from 5 up; $k = 5 \dots 18$ are the cases held here
V0 C1 optimality reviewed on these cases, second certificate, reported
Daniel after Burns, Massaccesi · evand square-packing 2026-10-03 · packet · packet · source · 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-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
2026-10-06 published T-104 · $n = 20$
Mixed rectangle-measure lower bound verified at $n = 20$ , on a declared net
V3 C3 lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-07 published T-124 · $n = 20, 21$
Reported non-strict local minima for 178 source configurations
V0 C0 restricted optimality recorded
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register

T-020: s(n)≥24/5=4.80s(n)≥24/5=4.80 for n=19,20,21n=19,20,21

The cases

Results on these cases

Links

T-020: $s (n) \geq 24 / 5 = 4.80$ for $n = 19, 20, 21$