Exact local exclusion; no new covering bound

Status: residue (2026-09-23).

Let $S$ be the 50-column set in ../H_r10_n50.txt. Every 49-element set obtained by removing two members of $S$ and inserting one nonzero vector has at least 9 uncovered syndromes. There is no covering set obtained by removing three members of $S$ and inserting two distinct nonzero vectors.

The exact target remains $\ell_2(10,2)\le49$. The prior certified record is $\ell_2(10,2)\le50$ from Wu, arXiv:2608.27494, also used by Davydov–Marcugini–Pambianco–Wu, arXiv:2609.16078. The earlier published Kaikkonen–Rosendahl construction gave 51 columns; see Davydov–Marcugini–Pambianco, arXiv:2511.02542, Table 5.1.

run_all.sh checks the source matrix, independently recounts the best two-delete/one-add candidate's nine holes, and reruns both complete local searches. Exit 0 means only the restricted predicates above hold. A two-delete/one-add candidate with fewer than nine holes, or a covering three-delete/two-add candidate, falsifies the claim. A 49-column covering set outside these neighborhoods would not falsify it and would establish the still-open bound.