Compatibility Restores Spin Alignment for Three Qubits

Abstract

We study a spectral majorization problem for lifted quantum marginals. For every three-qubit state, every single-qubit replacement state, arbitrary nonnegative weights on all eight supports, and compatible marginals arising from one global state, we prove that the weighted marginal sum is majorized by the perfectly aligned construction. This yields simultaneous bounds for all Schur-concave spectral entropies, together with a trace-norm robustness statement and a quantitative incompatibility witness. The proof isolates compatibility to the overlapping-pair sector and establishes unconditional bounds for the remaining extreme-ray families. We also prove several strict four-qubit slices, including endpoint and complement-symmetric prefixes, an equal-singleton rank-two ray, and a symmetric one-excitation Dicke family. The unrestricted noncommuting four-qubit middle-rank problem remains open. The deposited source package contains the exact certificates and reproducibility scripts.