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 ν_t(dλ) = e−λt dμ(λ) / F(t), we show that observables of the spectral variable satisfy the covariance law d/dt E_t[g(λ)] = −Cov_t(λ, g(λ)). In particular, the effective rate r(t) = −F′(t)/F(t) obeys the variance flow identity r′(t) = −Var_t(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 = Aδ_p + Bδ_q. The covariance law generates an infinite hierarchy of centered moment dynamics, m′k(t) = −m{k+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.