Spectral Selection under Exponential Tilting: Covariance Dynamics and Edge Asymptotics

We study the dynamical structure induced by exponential tilting of a positive measure μ on [0,∞) with finite second moment. The tilted family ν_t(dλ) ∝ e^{−λt} dμ(λ) satisfies the exact covariance identityd/dt E_t[g] = −Cov_t(λ,g),from which the variance dissipation lawr′(t) = −Var_t(λ) ≤ 0follows, where r(t) = E_t[λ].The family (ν_t) is a one-parameter exponential family whose Fisher information satisfiesI(t) = Var_t(λ) = −r′(t).This identity provides a closed scalar law governing spectral selection dynamics.We prove convergence r(t) → λ* = inf supp(μ) and classify rates: exponential under a spectral gap, and algebraic r(t) − λ* ~ β/t under regular variation μ([λ*,λ*+x]) ~ x^β L(x). We establish the global dissipation identity∫₀^∞ Var_t(λ) dt = r(0) − λ*,under a finite first-moment condition. For bi-atomic spectra, the dynamics closes exactly as a Riccati equation r′(t) = −(r−p)(q−r), saturating the variance bound.To our knowledge, the explicit formulation r′(t) = −I(t) as a standalone identity governing spectral selection does not appear explicitly in the literature.This framework provides a minimal and exact backbone for spectral flow, linking covariance dynamics, Fisher information, and asymptotic edge selection.