An Exact Law of Spectral Selection in Laplace Mixtures: Covariance Flow, Fisher Information, and Edge Asymptotics

We consider Laplace mixtures F (t) = ∫₀^∞ e^−λt dμ (λ) and the associated tilted family νₜ (dλ) ∝ e^−λt dμ (λ). The exponential tilting structure makes (νₜ) a one-parameter exponential family with natural parameter θ = −t and sufficient statistic λ. This structure yields two exact identities: the covariance evolution law d/dt Eₜg = −Covₜ (λ, g), and the Fisher information identity I (t) = Varₜ (λ) = −r′ (t), where r (t) = Eₜλ. These relations hold without approximation under a finite second-moment assumption. When combined with positivity and support conditions on μ, this structural core leads to edge selection r (t) → λ* = inf supp (μ). Under additional Tauberian hypotheses, convergence rates can be classified, including the asymptotic behavior r (t) − λ* ~ β/t under regular variation μ (λ*, λ* + x) ~ x^β L (x). The contribution is not in the individual components, which are classical, but in identifying exponential tilting as a common structural mechanism and organizing covariance dynamics, Fisher information, and asymptotic selection within a single exact framework.