Spectral Mean Flow of Laplace Mixtures: Variance Dissipation and Uniqueness of Finite Closure

We establish an exact dynamical law governing Laplace mixtures: r'(t) = -Var_t(λ), where r(t) is the effective decay rate under the tilted spectral measure. This identity defines an infinite-dimensional flow, which in general does not admit any finite-dimensional closure. We prove that under compact spectral support, the only possible quadratic closure of the variance is Var_t(λ) = (r - p)(q - r), and this closure uniquely forces the spectral measure to be bi-atomic: μ = A δ_p + B δ_q. This yields the Riccati equation r'(t) = -(r - p)(q - r), which characterizes the unique finite-dimensional regime of the spectral flow. This result provides a complete classification of spectral dynamics: - continuous spectra → non-closed dynamics- multi-mode discrete spectra → infinite hierarchy- two-mode spectra → exact Riccati closure (unique case) The support condition supp(μ) ⊂ [p,q] is essential for the rigidity result. This work forms the theoretical foundation for the COS45 framework, where the Riccati structure appears as the exact two-mode corollary of the spectral mean flow --- Note. This is a theoretical work. No claims are made regarding robustness under noise or direct applicability to empirical data.