Equality cases and every-copy results for the Werner endpoint inequality

Whether every entangled state with a non-positive partial transpose (NPT) is distillable reduces to the case of the Werner states ρ(d,α) = (I + αF)/(d² + αd), with F the swap, which for d ≥ 3 and −1/2 ≤ α < −1/d are NPT but one-copy undistillable. By the Schmidt-rank criterion of DiVincenzo et al. and Dür et al. (2000), as rewritten by P. Costa Rico (2025), ρ(d,α) is k-copy undistillable exactly when Σ_J α^|J| ‖Tr_J C‖² ≥ 0 for every vector of Schmidt rank at most two on k copies, written as a matrix C, where Tr_J traces out the copies in J. At α = −1/2 this inequality was proved independently for two copies by Fu et al., Fraser et al., Song and Chen, and Bharti et al. (2026), and for three copies by a computer-assisted proof in the companion record doi:10.5281/zenodo.23604343; if it held for every k, NPT bound entanglement would exist. We prove the inequality at α = −1/2 for every k and d ≥ 3 on the class of matrices C whose range is spanned by a vector f_0 of V_0 and a vector f_1 of V_1 and whose co-range (the range of C*) is arbitrary. Here V_c is spanned by |c⟩^⊗k and the k vectors obtained from it by replacing one factor with |2⟩, for orthonormal |0⟩, |1⟩, |2⟩. So no vector of this kind witnesses distillability of ρ(d,−1/2). We chose this class because, for generic f_0 and f_1, the projector onto the range has reduced states of rank three on every copy, while every known nonzero C with equality at α = −1/2 has a range with a reduced state of rank at most two on some copy. The proof reduces the claim by Schur complements to a quadratic form and then argues by induction on the number of copies. Python scripts in this record check the five rational identities of the proof symbolically; they do not involve k, and no other step uses a computer. We also prove that if the inequality holds for k − 1 copies but fails for k, then for some α > −1/2 it still holds, with equality at a nonzero C whose range has reduced states of rank at least three on every copy, as for generic members of the class. For two copies we prove that every nonzero C with equality at α = −1/2 has rank two and that the ranges of C and C* lie in products of two-dimensional single-copy subspaces. For general C the inequality remains open from four copies on.