T-021: $s (n) \geq 97 / 20 = 4.85$ for $n = 20, 21$

V3 C3 lower bound confirmed superseded by T-052 and T-104

2026-09-05 established · Levy after Burns, Massaccesi · $n = 20, 21$

$s (20) \geq 97 / 20$ and $s (21) \geq 97 / 20$ , from this project's weighted fractional unavoidable-set certificate at container side 97/20 = 4.85. Both sizes held 24/5 = 4.80 from T-020 earlier the same week in this register's verified fields; the movement is +0.05 at each. The certificate does not reach $n = 19$ : its total mass is 19848723/1000000 = 19.848723, above nineteen, so Condition 2 gives it $n = 20$ and upward and T-020's 24/5 rung, retained beside it, still carries $n = 19$ .

The two sizes take no monotonicity step, for the reason T-020 gives: only Condition 2 mentions n among the five conditions, so an atom set of mass M certifies its side for every integer strictly above M. From $n = 22$ on the register already holds 5, so the certificate is true there and weaker.

Where the rung came from is worth the record. A pre-registered bisection of [24/5, 9977/2000] ran four rungs unattended. The uniform grid walled at this side -- its restricted optimum crossed twenty at round 31 of 32 and stood at 20.000439 with placements still violated -- while a second site set, the 24/5 certificate's own 2260 atoms scaled by 97/96 and unioned with the grids, converged below twenty at the same side. The certificate is the second construction's; the first construction's crossing is why H-062 asks for two.

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 the same two sizes, one rung higher, and neither size changes character at 4.85. What it adds beyond a rung is a measurement rather than a theorem, and the measurement is registered as such: the m = 5 covering wall is bracketed to [97/20, 39/8] -- a certificate at 4.85, crossings above twenty on both constructions at 4.875 -- which is the first bracket this project has put around a covering wall, and it sits far below the method's ceiling of 4.9885 rather than at it. That bears on planning, not on the strength of this claim.
Composition: Primary at both sizes: 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-020 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 margin is 0.151277 below twenty, wider than T-020's 0.077380 was, so the side was not pushed to where the covering value stops it either. What is new is that the stopping side is now bracketed rather than guessed: 39/8 = 4.875 walls on both constructions and 97/20 certifies, so the m = 5 covering wall lies in an interval of width 0.025, and 979/200 = 4.895 and 997/200 = 4.985 wall as well. H-062 asked for 0.02 and the four decided rungs give 0.025, so the hypothesis is unresolved rather than accepted; the round is exp-061 and the remaining rung is the midpoint by the same rule.

One reading of the 4.985 wall does not survive inspection and is recorded here so nobody repeats it. Just below the ceiling the twenty-five axis-parallel B-squares of a 5 x 5 arrangement overlap only in strips of width 5B - L = 0.0035, and the restricted dual is checked at sites only, so a site set with no site in those strips makes twenty-five unit weights dual-feasible and the restricted optimum exactly 25.000000 whatever the covering value is. No uniform inset-1/2 grid at any count the lane could afford puts a site in every strip. The exactly round value this register has learned to distrust has, at m = 5, a mechanism.

The $n = 21$ half of the margin was taken on 2026-09-23: T-034 certifies 122/25 = 4.88 at $n = 21$ from an unseeded site set and supersedes this result there. What this result still held alone was $n = 20$ , whose bound stayed 97/20 because T-034's atoms are too heavy for it, until wand125's replayed rectangle certificate, $s (20) \geq 979 / 200$ (T-045), superseded it there on 2026-10-02. This result stays true as stated at both of its sizes.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps

The cases

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

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

2005 published T-007 · $n = 20, 21$
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
V0 C1 lower bound incomplete on these cases, superseded by T-052 and T-104
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 established T-020 · $n = 20, 21$
$s (n) \geq 24 / 5 = 4.80$ for $n = 19, 20, 21$
V3 C3 lower bound confirmed superseded by T-052 and T-104
Levy after Burns, Massaccesi · register
2026-09-04 published T-085 · $n = 20, 21$
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 this result · $n = 20, 21$
$s (n) \geq 97 / 20 = 4.85$ 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 · $n = 20$
Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$
V3 C3 lower bound confirmed superseded by 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 · $n = 20$
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
V0 C0 lower bound recorded superseded by 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 · $n = 20, 21$
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 · $n = 20, 21$
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
V3 C3 lower bound confirmed on these cases, superseded by T-052 and T-104
Karakuş · Karakuş 2026 · source · register
2026-10-01 published T-068 · $n = 20$
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
V3 C3 lower bound confirmed on these cases, superseded by T-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 · $n = 20$
Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$
V3 C3 lower bound confirmed on these cases, superseded by T-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-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-021: s(n)≥97/20=4.85s(n)≥97/20=4.85 for n=20,21n=20,21

The cases

Results on these cases

Links

T-021: $s (n) \geq 97 / 20 = 4.85$ for $n = 20, 21$