Primitive Spectral Coordinates for Positive Definite Density Matrices

We specialize Zimmerman's primitive-sequence formalism for probability measures on compact intervals to the empirical spectral distributions of finite-dimensional density matrices. Exploiting finite support and the unit-trace constraint, we prove that the first N-1 primitive coordinates uniquely determine the spectrum of an N×N positive definite density matrix, and therefore all polynomial spectral invariants. Restricted to the regular stratum of simple spectrum, these quantities define a smooth local coordinate system on the regular spectral quotient. The resulting framework provides an alternative smooth coordinate realization of mixed-state spectral geometry through repeated antidifferentiation of cumulative spectral distributions, establishing a bridge between Lie--Poisson geometry and iterative integration.