Spectral Selection under Exponential Tilting: Covariance Dynamics and Edge Asymptotics

We study the dynamical structure induced by exponential tilting of a positive measure mu on 0, infinity) with finite second moment. The tilted family nuₜ (dlambda) proportional to e^-lambda t dmu (lambda) satisfies the exact covariance identity d/dt Eₜ[g = -Covₜ (lambda, g), from which the variance dissipation law r' (t) = -Varₜ (lambda) lambda_* = inf supp (mu) and classify convergence rates: exponential under a spectral gap, algebraic of order beta/t under regular variation mu (lambda_*, lambda_*+x) ~ xᵇeta L (x). The global identity integral₀ⁱnfinity Varₜ (lambda) dt = r (0) - lambda_* holds under a finite first-moment condition. For a two-point spectrum mu = A deltaₚ + B deltaq, the dynamics reduces to the exact Riccati equation r' (t) = - (r-p) (q-r), which saturates the variance bound. To our knowledge, the explicit formulation r' (t) = -I (t) as a named result governing spectral selection does not appear in the existing literature on gradient flows, Markov semigroups, or entropy dissipation.