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 nu_t(dlambda) proportional to e-lambda t dmu(lambda) satisfies the exact covariance identity d/dt E_t[g] = -Cov_t(lambda, g), from which the variance dissipation law r'(t) = -Var_t(lambda) <= 0 follows. The family (nu_t) is a one-parameter exponential family; its Fisher information coincides with Var_t(lambda), giving the identity -r'(t) = I(t). We prove convergence r(t) -> 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]) ~ x^beta L(x). The global identity integral_0^infinity Var_t(lambda) dt = r(0) - lambda_* holds under a finite first-moment condition. For a two-point spectrum mu = A delta_p + B delta_q, 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.