Keep any m of the qd edge coordinates and fix all others to zero. Restrict each vertex parity check to the retained coordinates; the resulting matrix is A0. Every vector in its kernel extends by zeros to an accepted full edge word, so nonzero weight remains at least qd/P>=m/P. Merely deleting bits without fixing them would not justify this argument. Each row has at most d<=D0 nonzeros. There are 2q ceil(d/(8B)) rows. If q=1, d=m gives at most n/4+2<=3n. If q>1, d=D0>=8B gives fewer than m/B<=n rows. This completes the stated initial-check contract.