We establish an exact dynamical law governing Laplace mixtures: r' (t) = -Varₜ (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 Varₜ (λ) = (r - p) (q - r), and this closure uniquely forces the spectral measure to be bi-atomic: μ = A δₚ + B δq. 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.
Louis Morissette (Thu,) studied this question.