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 ν_t(dλ) ∝ e^{−λt} dμ(λ). The exponential tilting structure makes (ν_t) 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 E_t[g] = −Cov_t(λ, g), and the Fisher information identity I(t) = Var_t(λ) = −r′(t), where r(t) = E_t[λ]. 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.