COS 45 – Riccati Dynamics of Effective Rates in Exponential Mixtures

[EN] This work establishes the COS 45 crossover as an exact analytic corollary of the quadratic spectral closure theorem (Spectral Mean Flow v3.1). Starting from the rigidity lemma — which forces any exponential mixture whose variance closes quadratically to be bi-atomic — we derive the exact Riccati dynamics of the effective rate, its projective Möbius representation, and the parallel structure for power-law mixtures. In the normalized regime p=1/2, q=1, the crossover α(δ)=1/√2 occurs at δ*=a²/(2b²), defining COS 45 as an exact analytic marker of the two-mode transition, not a universal constant. [FR] Ce travail établit le croisement COS 45 comme corollaire analytique exact du théorème de fermeture spectrale…