Anti-Blocking Codegrees, Stable-Set Polytopes, and Prime-Factorization Polytopes

We prove a codegree theorem for lattice polytopes obtained from finite coordinate-down-closed subsets of Z^A≥ 0. The theorem identifies the Ehrhart codegree with a blocker optimum and, when an anchor set is present, splits that optimum by an explicit affine decomposition. This gives a single mechanism behind two models: stable-set polytopes and prime-factorization polytopes. For every finite simple graph G, with the usual empty-graph convention, codeg(P_G) = floor(chi_f(G)) + 1, deg h^*P_G(t) = |V(G)| - floor(chi_f(G)). For the prime-factorization polytope P_m, the large primes form an anchor set. If G_m is the bipartite graph between small primes q <= sqrt(m) and large primes p > sqrt(m), with edge qp <= m, and U(m) is its Hall deficiency, then for every m >= 10, pi(sqrt(m)) - deg h^*P_m(t) = floor(U(m)/2). This is a preprint version.