Certified bound

The exact claim is

The published Table 7.2 bound of Davydov–Marcugini–Pambianco, arXiv:2511.02542v1, is 26238. The notebook previously certified 26206. Thus this construction saves 63 columns against the published table and 31 against the prior notebook bound.

The seed is the certified $26\times817$ radius-3 matrix. Its inherited 34-block $(3,0)$-partition was established in q13. An exact search of all 561 unordered block pairs found two merges preserving the partition: $(7,31)$ and $(15,24)$ (zero-based labels). The committed partition uses $(7,31)$ and has 33 blocks. The independent C verifier sweeps all $2^{26}=67{,}108{,}864$ seed syndromes under those labels and finds all of them covered by at most three columns from distinct blocks.

Construction QM$_4^3$ (Theorem 6.1, (6.4), (6.8)) applies with $m=5$ since $817\ge2^5+1=33$. The indicator set is $\mathbb F_{32}\cup\{*\}$, with $\mathbb F_{32}=\mathbb F_2[x]/(x^5+x^2+1)$. It yields

The verifier checks the field, rank 41, 26,175 distinct nonzero columns, and every $D_3$ or $A(h,\beta)$ column identity. The theorem supplies the output covering implication; no $2^{41}$ flat sweep is claimed. The radius is exactly 3 by the radius-2 sphere-volume bound.

Replay from problems/covering/:

sh compute/q16/run_all.sh

Falsifiers: a missing seed syndrome under the 33-block labels, invalid field arithmetic or indicator assignment, rank below 41, a wrong output column, or an uncovered output syndrome. No optimality claim.