Spectral Mean Flow in Laplace Mixtures — Variance Dissipation, Replicator Dynamics, and Riccati Closure

We study the dynamical structure induced by Laplace mixtures of exponential modes, F(t)=∫e−λt dμ(λ),F(t)=∫e−λtdμ(λ), where μμ is a positive measure with finite first moment. Introducing the normalized tilted spectral measure νt(dλ)=e−λt dμ(λ)F(t),νt(dλ)=F(t)e−λtdμ(λ), we show that observables of the spectral variable satisfy the covariance law ddt Et[g(λ)]=−Covt(λ, g(λ)).dtdEt[g(λ)]=−Covt(λ,g(λ)). In particular, the effective rate r(t)=−F′(t)/F(t)r(t)=−F′(t)/F(t) obeys the variance flow identity r′(t)=−Vart(λ)≤0,r′(t)=−Vart(λ)≤0, revealing a dissipative dynamics on the spectral distribution. The special case of two exponential modes yields an exact autonomous Riccati equation r′(t)=−(r−p)(q−r)r′(t)=−(r−p)(q−r), which is the unique quadratic closure of the moment hierarchy. These results show that Laplace mixtures naturally carry a covariance-driven spectral dynamics, linking Laplace transform theory with dissipative flows on probability measures.