Spectral Mean Flow and Variance Dynamics of Completely Monotone Functions

We study the spectral mean r(t) and cumulant dynamics of completely monotone functions via the Bernstein–Widder framework. We establish four main results: a universal variance identity r'(t) = −Var(λ) ≤ 0; a four-way equivalence characterising bi-atomicity via Riccati dynamics; an exact cumulant cascade κₙ'(t) = −κₙ₊₁(t); and an exhaustive classification of polynomial autonomous closures. The spectral variance is identified as a binomial-class QVF, with algebraic exclusion of the Gamma family.