Sharp Plucker Geometry for Three-Copy Werner Distillation

Whether negative-partial-transpose entanglement can remain undistillable is a longstanding problem in quantum information theory. We analyze the first unresolved three-copy Werner endpoint using a sharp dimension-free inequality for complementary partial traces and an optimal exterior-square inequality for orthonormal tripartite vectors. The latter identifies local SWAP- parity statistics with metric data of a decomposable Plucker bivector. Together these inequalities prove endpoint nonnegativity for every positive semidefinite rank-two coefficient operator and for the complete normal rank-two sector in arbitrary finite local dimensions. For genuinely nonnormal operators, an exact crossed-Gram criterion proves nonnegativity when one local outpu-input support overlap is at most two, including every system with a qubit-sized factor, and when either support plane contains a product ray. Explicit anti-state and rank-boundary families establish optimality of the constants and the rank restriction.