Spectral Uniqueness and Gauge Ambiguity in Quantum Mereology

This work separates two notions of uniqueness that are often conflated in spectral approaches to quantum subsystem reconstruction. For a fixed bipartite dimension (D=mn), Hamiltonian eigenvalues are placed on an (m× n) grid and scored by their least-squares distance from an additive table (Eᵢⱼ=a_i+b_j). It is proved that, for Lebesgue-almost every spectrum, the optimal spectral labeling is unique modulo row and column permutations and, for square grids, factor exchange. The exact non-uniqueness locus is a semialgebraic subset of a finite union of proper quadratic hypersurfaces. A positive selection gap additionally provides an explicit perturbation bound under which the optimal labeling remains unchanged. This generic spectral uniqueness does not determine a unique embedded tensor-product structure. For a nondegenerate Hamiltonian and a fixed labeling, the residual centralizer orbit is classified exactly as[ T⁽ᵐ⁻¹⁾⁽ⁿ⁻¹⁾.]Its complete invariants are gauge-invariant plaquette phases. The additive spectral residual and the phase ambiguity are shown to arise from the same cycle structure of the complete bipartite graph (Km,n): the former is its real cycle space, while the latter is its (U(1)) character torus. Finally, continuous operational anchoring is shown to require at least ((m-1)(n-1)) real response coordinates for local identification, and strictly more for global identification.