The Two-Copy Purity Body and the Complementary-Block Frontier for Werner Distillability

For an operator R of rank at most two on C^d ⊗ C^d, the four subset purities ‖R‖_HS², ‖Tr₁ R‖_HS², ‖Tr₂ R‖_HS², |Tr R|² satisfy the sharp two-copy Werner inequality proved in July 2026 and the rank bound |Tr R|² ≤ 2‖R‖_HS². We determine every linear inequality they satisfy in unrestricted local dimension and in every even dimension. In unrestricted local dimension the closed convex hull of the normalized purity vectors is a polytope with six vertices and seven facets; two of the facets are compression inequalities that the earlier bounds do not imply. In each even local dimension d the body is that polytope cut by two Cauchy–Schwarz inequalities, with vertices depending on d; the same cut is not the body in any odd dimension, because a vertex it predicts is not the purity vector of any rank-two operator. At r copies, the certificates built from valid two-block inequalities on complementary groupings of the sites, the family used by Bharti, Gajjala and Haug, reach exactly the interval they found: an explicit dual vector shows that no combination of dimension-free two-copy inequalities on complementary blocks does better, at any copy number. The compression inequalities come from averaging the one-copy inequality over random compressions of one site; the dual vectors are assembled from vertices of the two-copy body, so the body decides the frontier.