The Spectral Covariance Law and Edge Selection in Laplace Mixtures: Exact Identities and Dissipation

We establish the exact mathematical backbone governing Laplace mixtures of the form F(t) = ∫ e-λt dμ(λ), under explicit admissibility conditions. No approximation is introduced. We prove that the spectral center r(t) = E_t[λ] satisfies the exact identity r′(t) = −Var_t(λ), yielding a dissipation law ∫₀^∞ Var_t(λ) dt = r(0) − λ*, where λ* is the lower spectral edge. All results are derived under minimal assumptions (μ ≥ 0, supp(μ) ⊂ [0,∞), finite second moment). Edge asymptotics are cited from regular variation theory. This document contains only exact identities. It does not address detection, decision, or parameter reconstruction. Global pole reconstruction remains open. This framework does not impose a result. It measures when a result is justified.