Impose the same local check matrix on the port-ordered edge bits at every left and right vertex. Consider a nonzero accepted edge word of weight w. Let U and W be its touched vertices. Local distance forces w>=delta d max(|U|,|W|)>=delta d sqrt(|U||W|). All its one-edges lie between U and W, so w<=e(U,W). Combining with the mixing bound and dividing by d sqrt(|U||W|)>0 gives sqrt(|U||W|)>=(delta-epsilon)q. Therefore w>=delta(delta-epsilon)qd=qd/P, since epsilon=delta/2. Local constraints alone do not supply this global distance; the graph mixing bound is also necessary.