Certified bound

Status: certified dent (2026-09-11).

The target inequality is

\ell_2(44,3)\le 52351.

The prior published upper bound is 52415 in Davydov–Marcugini–Pambianco, arXiv:2511.02542v1, Table 7.2. The improvement is 64 columns. This is a continuation of the notebook's existing 817-column code, not a new small seed.

The 34-block $(3,0)$ -partition of the $26$ by $817$ seed is inherited from its QM₅³ construction: 23 Golay $A$ -blocks, the 10 blocks of the embedded $10$ by $50$ radius-2 matrix, and one W₅ block. The committed refinement has 65 nonempty blocks. Its block 0 receives the indicator $*$ , and blocks $1,\ldots,64$ receive the elements $0,\ldots,63$ of the field

\mathbb F_{64}=\mathbb F_2[x]/(x^6+x+1).

Thus Construction QM $_4^3$ (Theorem 6.1, (6.4), (6.8)) applies with $m=6$ :

817\ge 2^6+1=65=p(H_0,0),\qquad r=26+3\cdot6=44,

and

n=2^6(817+1)-1=52351.

verify_q13.c is independent of the Python builder. With one 8 MiB bitset it exhausts all $2^{26}=67{,}108{,}864$ seed syndromes under both partitions. The 34-block partition covers 321,037 syndromes with at most two columns and all 67,108,864 with at most three; the 65-block refinement covers 321,067 and all 67,108,864, respectively. The same verifier checks the field, refinement, input and output ranks, distinct nonzero columns, and every one of the 52,351 $D_3$ or $A(h,\beta)$ output-column identities. Theorem 6.1 then certifies output covering; there is no $2^{44}$ sweep. The radius is exactly 3 because $1+n+\binom n2<2^{44}$ .

Replay from problems/covering/:

compute/q13/run_all.sh

Falsifiers: a missing syndrome in the seed partition, an invalid indicator assignment or field, an incorrect emitted column, rank below 44, or an uncovered syndrome of the output. No optimality claim.