Covariance Dynamics and Low-Rank Structure in Laplace Spectral Mixtures

We study Laplace mixtures F(t)=∫0∞e−λt dμ(λ),F(t)=∫0∞e−λtdμ(λ), under minimal regularity assumptions. The effective rate r(t)=∫λe−λtdμ∫e−λtdμr(t)=∫e−λtdμ∫λe−λtdμ satisfies the exact identity r′(t)=−Vart(λ),r′(t)=−Vart(λ), which defines a dissipative infinite-dimensional flow. In the bi-atomic case, this dynamics admits a closed Riccati form r′(t)=−(r−λ1)(λ2−r),r′(t)=−(r−λ1)(λ2−r), which is equivalent to a representation of FF as a sum of two exponentials and implies Hankel rank at most two in the noiseless discrete setting. We analyze the structural relations between covariance dynamics, finite-dimensional closure, and low-rank Hankel representations. We also discuss heuristic detectability criteria based on the third Hankel singular value under noise, emphasizing their non-universal character. This work isolates an exact structural backbone linking spectral dynamics, curvature, and low-rank observability, while leaving statistical estimation and higher-order mixtures as open problems.