Spectral Mean Flow of Laplace Mixtures: Variance Dissipation, Moment Cascade, and Riccati Closure

We study the dynamical structure induced by Laplace mixtures of exponential modes, F (t) = ∫ e^−λt dμ (λ), where μ is a positive measure with finite first moment. Introducing the normalized tilted spectral measure νₜ (dλ) = e^−λt dμ (λ) / F (t), we show that observables of the spectral variable satisfy the covariance law d/dt Eₜg (λ) = −Covₜ (λ, g (λ) ). In particular, the effective rate r (t) = −F′ (t) /F (t) obeys the variance flow identity r′ (t) = −Varₜ (λ) ≤ 0, revealing a dissipative dynamics on the spectral distribution. This implies monotone decrease of the effective rate and asymptotic dominance of the slowest spectral mode p = inf supp (μ). The special case of two exponential modes, F (t) = Ae^−pt + Be^−qt, yields an exact autonomous Riccati equation r′ (t) = − (r−p) (q−r), the quadratic closure associated with a bi-atomic spectral measure. We prove that this closure characterizes bi-atomic spectral measures: quadratic variance closure forces μ = Aδₚ + Bδq. The covariance law generates an infinite hierarchy of centered moment dynamics, m′k (t) = −mk+1 (t), in which each level drives the next. This moment cascade does not close unless the spectral measure has finite support; in the bi-atomic case it closes at order two, recovering the Riccati equation. These results show that Laplace mixtures naturally carry a covariance-driven spectral dynamics, linking Laplace transform theory with dissipative flows on probability measures.