T-034: $s (21) \geq 122 / 25 = 4.88$

V3 C3 lower bound confirmed superseded by T-052

2026-09-23 established · Levy after Burns, Massaccesi · $n = 21$

$s (21) \geq 122 / 25$ , from this project's weighted fractional unavoidable-set certificate at container side 122/25 = 4.88. The case held 97/20 = 4.85 from T-021 in this register's verified field, and the movement is +0.03. The certificate does not reach $n = 20$ : its total mass is 5036431/250000 = 20.145724, above twenty, so Condition 2 gives it $n = 21$ and upward, and T-021's 97/20 rung, retained beside it, still carries $n = 20$ .

The size takes 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.

The site set is the plainest the generator offers: auto grids (34, 46, 56) at inset 1/2 with no seed and no windows, converged below twenty-one in 36 LP rounds. At the same side a T-021-seeded set ran out its deadline unconverged, and a five-window set converged but stalled the interval route on a seam where atom rows sit exactly B apart; neither run decides anything about the side.

Significance, composition and next rung

Significance: Scored at S3 by the calibration note T-020 wrote and T-021 applied: "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 one of the same sizes, one rung higher, and $n = 21$ does not change character at 4.88. What it adds beyond the rung is a reading about the instrument rather than a theorem: the stock generator with no seed and no windows converged where the seeded and windowed site sets did not decide. The side also sits above 39/8 = 4.875, where both of H-062's constructions crossed twenty at $n = 20$ ; a mass above twenty is consistent with that wall and says nothing about $n = 20$ .
Composition: Primary at $n = 21$ : 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, as for T-021; that is C3, with the two methods shown beside the rung. The standard-library verifier the review also ran is a second implementation of the sweep's method, so it adds nothing to that count.

One adversarial AI review is retained, the mapped 2026-09-23 review by Fable at max thinking, which covers the complete claim: it re-decided the frozen bytes by three routes, checked each of the five conditions and the $n = 21$ -only scope, and found nothing that refutes them. A different agent than the lane that produced the certificate wrote it, inside the same session; it is not an external review, and it is not a human oversight record.
Next rung: The margin is 0.854276 below twenty-one, more than five times T-021's 0.151277 below twenty, so the side was not pushed to where the covering value stops it. H-240 froze it below X-047's estimated additive crossing near 4.886, which is an estimate and not a bound, and this run did not test it.

A certificate for $n = 21$ cannot exist above ceil(sqrt(21)) * B = 4.9885, leaving 0.1085 of runway above 122/25, and the best known packing is the trivial 5, so the ceiling binds first: the method could in principle close the case to within 0.0115 of the upper bound and can never reach it.

The next rung is a pre-registered side above 122/25 on the same unseeded construction, read from the restricted optimum a converged run reaches. Set B's refusal is the instrument note to carry with it: a window lattice at pitch B puts atom rows exactly B apart, and the interval route stalls on the seam where an upright square has both closed edges on such a pair. A stall there is not a counterexample, and it decides nothing.

V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record; rung 5 would need a proof-assistant formalization of the five-condition theorem, reviewed by human experts.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps

The case

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 the case

12 results in the register on $n = 21$ , 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-052
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 established T-020
$s (n) \geq 24 / 5 = 4.80$ for $n = 19, 20, 21$
V3 C3 lower bound confirmed superseded by T-052
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
2026-09-05 established T-021
$s (n) \geq 97 / 20 = 4.85$ for $n = 20, 21$
V3 C3 lower bound confirmed superseded by T-052
Levy after Burns, Massaccesi · register
2026-09-23 established T-034 this result
$s (21) \geq 122 / 25 = 4.88$
V3 C3 lower bound confirmed superseded by T-052
Levy after Burns, Massaccesi · register
2026-09-23 published T-050
$s (21) \geq 5000 / 1001 = 4.995004995 \dots$
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-052
$s (21) = 5$ , 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
$s (21) = 5$ 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
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-052
Karakuş · Karakuş 2026 · source · register
2026-10-03 published T-081
$s (k^{2} - 4) = k$ for every integer $k$ from 5 up; $k = 5 \dots 18$ are the cases held here
V0 C1 optimality reviewed on this case, second certificate, reported
Daniel after Burns, Massaccesi · evand square-packing 2026-10-03 · packet · packet · source · review · register
2026-10-07 published T-124
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

Links

On this site: Case record, $n = 21$ · Frontier row, $n = 21$ · T-034 in the results table

On GitHub, at main

Register: T-034 in results.yaml, line 2890
Evidence: E-n021-fractional-certificate-122-25 · E-fractional-interval-decision
Proofs and certificates: certificate certificate.json · certificate certificate.json
Sources: Burns–Massaccesi n17
Artifacts: 7 artifacts and controls
cases/n21_fractional_certificate/certificate.json campaign/series/series-000-smoke-and-calibration/results/agenda-042/exp-229-n21-122-25-receipt.md campaign/series/series-000-smoke-and-calibration/results/agenda-042/exp-229-n21-C-family.json devtools/run_fractional_colgen.py src/sqpack/fractional/colgen.py tests/test_fractional_certificate.py tests/test_fractional_interval.py
Case file: frontier/n-021.md (verified lower, verified upper, reported lower, reported upper)

T-034: s(21)≥122/25=4.88s(21)≥122/25=4.88

The case

Results on the case

Links

T-034: $s (21) \geq 122 / 25 = 4.88$