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 ZA 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 (PG) = floor (chif (G) ) + 1, deg h^*₏₆ (t) = |V (G) | - floor (chif (G) ). For the prime-factorization polytope Pₘ, the large primes form an anchor set. If Gₘ is the bipartite graph between small primes q sqrt (m), with edge qp = 10, pi (sqrt (m) ) - deg h^*₏䂷 (t) = floor (U (m) /2). This is a preprint version.