COS 45 as a Two-Mode Corollary of the Spectral Mean Flow: Riccati Dynamics and Structural Rigidity

COS45 is not an independent law. It is the exact two-mode corollary of the Spectral Mean Flow identity r'(t) = -Var_t(λ), which governs Laplace mixtures under the tilted spectral measure. We show that, under compact spectral support, the only possible quadratic closure of the variance is Var_t(λ) = (r - p)(q - r), and that this closure uniquely forces the spectral measure to be bi-atomic: μ = A δ_p + B δ_q. This yields the Riccati equation r'(t) = -(r - p)(q - r), establishing an exact equivalence between bi-atomic spectra, quadratic variance closure, and Riccati dynamics. In this regime, the dynamics admits an exact Möbius linearization, and the normalized midpoint α = 1/√2 emerges as a geometric invariant of the two-mode transition (COS45). This provides a falsifiable structural signature of two-mode spectral organization:any deviation from the Riccati dynamics indicates additional spectral modes or continuous structure. The support condition supp(μ) ⊂ [p,q] is essential for the rigidity result. --- Note. This is a theoretical work. No claims are made regarding robustness under noise or direct applicability to empirical data.