Hankel Rigidity and Rank Detection in Laplace Mixtures: A Conditional Bridge to Spectral Closure

We study the relation between Hankel operators and spectral structure in Laplace mixtures of the form F (t) = ∫ e^-λt dμ (λ). We show that the third singular value of the truncated Hankel operator provides a quantitative measure of deviation from the two-mode exponential model. We establish a conditional equivalence between three quantities: - the Hankel defect σ3 (HF), - the distance to the rank-2 model dist (F, V2), - the deviation from Riccati dynamics arising from the Spectral Mean Flow r' (t) = -Varₜ (λ). This provides a bridge between spectral dynamics, algebraic structure, and observable diagnostics. However, this equivalence holds only under the following conditions: - spectral gap, - transverse injectivity, - sufficient resolution. Without these conditions, the equivalence may fail. Therefore, Hankel structure provides a conditional observable for detecting two-mode spectral organization, but does not yield a global characterization. --- Note. This is a theoretical work. No claims are made regarding robustness under noise or direct applicability to empirical data.