X. Spectral and Dirichlet-Series Representations of Modular Survivor Interfaces

We associate with a finite symmetric affine prime system an exact periodic survivor observable on a primorial CRT torus. Its local factors are Yₚ (u, H) =p-2aₚ (H) +rhoₚ (u;H), where rhoₚ is the residue-overlap profile of the local obstruction cloud. We prove a finite Fourier factorization and an exact Dirichlet-series representation as a finite Hurwitz-zeta combination. We then study the diagonal von Mangoldt sum sum₍≤ₗ RP (n, H) Lambda (n), where Lambda is applied to the diagonal parameter, not to the affine forms. For z≤A log log X, a reduced-mean Euler product and the Siegel--Walfisz theorem give an unconditional low-level asymptotic under finite reduced local nonvanishing. Under GRH for Dirichlet L-functions, the same argument gives a pointwise extension to Qᵦ≤X^1/2-epsilon. The normalization is calibrated on the two-point pattern 0, 2, recovering the finite products converging to 2C₂. A Liouville parity-balance theorem shows that low-level survivor masks remain factorization-parity blind on average.