VIII / IX Finite Local Factors and Low-Level Prime Mass on Survivor-Restricted Additive Fibres

We study finite local obstruction factors attached to degree-one integer polynomial systems and additive survivor fibres. For a finite system of affine forms, viewed as degree-one polynomials, we define the local obstruction count nu_p and the corresponding finite Bateman-Horn/Dickson local factor. This finite factor is exact and is realized as a CRT survivor density. We then specialize to constant-sum fibres x+y=N and prove an unconditional low-level prime-mass theorem for the one-coordinate von Mangoldt weight under two-coordinate survivor restrictions. In the range Q_z <= (log M)^A, the Siegel-Walfisz theorem gives a uniform asymptotic for central fibres; under GRH for Dirichlet L-functions the same decomposition gives a conditional extension to Q_z <= M^(1/2-epsilon). For r>=3, we compute exact local factors for r-fold additive survivor fibres, prove convergence of the finite singular factors, and identify the r=3 limit with the ternary Goldbach singular series. The paper isolates finite local factors and low-level global realizations available from classical analytic input; it does not prove the Hardy-Littlewood or Bateman-Horn conjectures.