Mellin Transform, Height Zeta Function, and the DigammaGamma Bridge to AΦ(t)

We establish three results connecting the constitutive function Γ^Φ(x) of a simply-laced root system to the universal operator A_Φ(t). First, the Mellin transform of Γ^Φ factorises as 𝓜Γ^Φ = 𝓜Γ̃ · Z^Φ(s), where 𝓜Γ̃ = −Γ(s)[s(s+1)cos(πs/2) + 2sin(πs/2)] is a universal factor independent of the root system, and Z^Φ(s) = Σ_{k=1}^{h−1} m_k k^{−s} is a finite Dirichlet series encoding the height distribution {m_k}. Second, via the digamma recursion ψ(z+1) = ψ(z) + 1/z, we derive the spectral identity Σ_{k=1}^{h−1} 1/(t²+k²) = (1/2it)[ψ(h+it) − ψ(1+it) − ψ(h−it) + ψ(1−it)], which connects the spectral sum directly to the logarithmic derivative of the gamma ratio Γ(h+it)/Γ(1+it) sealed in R91. Third, substituting into ∂_t log A_Φ(t) = 2/t + 4t Σ 1/(t²+k²) and integrating, we recover exactly the gamma form A_Φ(t) = t² |Γ(h+it)/Γ(1+it)|⁴ of R90 with integration constant C = 1, closing the bridge Γ^Φ → A_Φ without intermediate combinatorial construction. The factorisation makes explicit the level separation established in R91: A_Φ depends only on the Coxeter number h, while Z^Φ carries the full arithmetic of the height distribution.