Operator-Norm Factor Leakage: An Exact Global Solution for Two-Qubit Tensor-Product Selection

We formulate tensor-product selection as a dynamical closure problem for full matrix-factor algebras. For a Hermitian operator H ∈ M₄(C) and a two-qubit factor pair (A,B), let Λ_H(A) be the worst-case operator-norm component of i[H,X] that leaves A, with X ranging over the self-adjoint unit ball of A. Let d_dec(H;A,B) be the operator-norm distance from H to the local-sum space A_sa + B_sa. We prove the exact fixed-factor identity Λ_H(A) = Λ_H(B) = 2 d_dec(H;A,B). We then optimize over all 2×2 tensor-product structures. If λ₁ ≤ λ₂ ≤ λ₃ ≤ λ₄ are the eigenvalues of H, the global minimum is one half of the least additive-rectangle defect of the spectrum. For a simple spectrum this becomes min_(A,B) max{Λ_H(A),Λ_H(B)} = ½ |(λ₄−λ₃)−(λ₂−λ₁)|. The proof gives more than the value: the closest local-sum Hamiltonian is unique for a simple spectrum, and all minimizing tensor-product structures form one orbit under the centralizer of H, after local basis changes and factor exchange are removed. A complementary stable-carrier theorem classifies the largest zero-leakage subalgebra for pointer-block Hamiltonians. The result therefore solves a specified worst-case, infinitesimal two-qubit selection problem; it does not assert that a Hamiltonian determines an absolute subsystem structure independently of the chosen functional and of additional physical anchors.