Grouped endpoint forms and a sharp interior law for the Werner distillability inequalities

Preprint and verification scripts for the paper "Grouped endpoint forms and a sharp interior law for the Werner distillability inequalities". For an operator C on H_1 ⊗ … ⊗ H_k, let q_k(−t,C) = Σ_J (−t)^{|J|} ‖Tr_J C‖², over sets J of factors. For rank at most two, nonnegativity of q_k(−1/2,C) is the Schmidt-rank-two test of undistillability of a Werner state from k copies; for k ≥ 4 it is open. We prove an identity, for every k, t and operator, writing q_k(−t,C) − (1−2t)(1−t)^{k−1} ‖C‖² as a combination, with explicit weights nonnegative on [0,1/2], of the endpoint forms of all factor groupings; if a factor has dimension at least two, no other constant admits such an identity with C-independent weights. So, over all local dimensions and for rank at most two, the sharp bound q_k(−t,C) ≥ (1−2t)(1−t)^{k−1} ‖C‖² on [0,1/2] is equivalent to the endpoint inequality; for k = 4, given the computer-assisted three-copy endpoint, it holds for a fixed C if and only if q_4(−1/2,C) ≥ 0. The sharp bound holds for every k when the range or co-range of C contains a fully product vector, and the identity extends to every real m ≥ 1, with 1/m replacing 1/2. For operators with range and co-range in the strings with exactly one digit 2, we give an identity and pair bounds for every k, prove the endpoint inequality for every k on symmetric sectors and, for k = 4, on further classes, one of them given the three-copy endpoint. The inequality also holds when the range lies in the span of the vectors |j⟩^{⊗k} (for even k at every rank), and for k = 4 at every rank when the range lies in a three-dimensional span of W-type vectors. Files: the preprint (PDF); ten Python scripts that check the explicit weights, identities and examples of Sections 3, 4 and 6–8 (exact rational arithmetic, except the checks of Section 7 and of the operator C_0 in Proposition 8.3, which use floating point); README.md, which says what each script checks and how to run it; MANIFEST.tsv with the size and SHA-256 of every file.