Spectral Correspondence Between ΓΦ and AΦ: The Generating Function FΦ(z)

We establish a structural identity connecting the analytic object Γ_Φ(x) of the TCFQ programme with the constitutive operator A_Φ(t) for every irreducible simply-laced root system Φ. The central result is the decomposition F_Φ(z) = Σ_{k=1}^{h-1} mₖ F_A₁ (zᵏ), where F_Φ is the spectral generating function of Γ_Φ and F_A₁ is an explicit closed-form function with a single logarithmic singularity. The logarithmic singularities of F_Φ at zₖ = −1 are in bijective correspondence with the factors (t² + k²)² of A_Φ(t), with logarithmic weight 2mₖ. Verification is recorded for A₁ and A₂.