Spectral Mean Flow and Variance Dynamics of Completely Monotone Functions

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━[English]━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━We study functions representable as finite positive mixtures of exponential modes, F(t) = Σ aₖ exp(−λₖt), and their associated effective rate r(t) = −F′(t)/F(t). We prove three exact results: a universal variance identity r′(t) = −Varₜ(λ) ≤ 0; a bi-exponential Riccati closure r′(t) = −(r−p)(q−r) which characterizes observables of the form Ae^{−pt} + Be^{−qt}; and a cumulant hierarchy r^(n)(t) = (−1)ⁿ κₙ₊₁(t). The Riccati dynamics is the projective image of a simple exponential flow in a hidden coordinate z(t). This provides a dynamical interpretation of hazard-rate evolution in exponential mixtures.…