Gamma-factor parity and pole removal in smooth zero-counting constants for zeta, beta, and the Gaussian number field

This work studies the additive constants that appear in smooth upper-half zero-counting main terms for the completed Riemann zeta function, the completed Dirichlet beta function, and the completed Dedekind zeta function of the Gaussian number field Q(i). Under one fixed upper-half counting convention, the constants split into two algebraic sources: the parity of the gamma factor and the contribution of the pole-removing factor. The paper computes these contributions uniformly. For a primitive Dirichlet character chi, the gamma-and-pole contribution is -chi(-1)/8 plus the pole contribution only in the principal conductor-1 case. For a number field with r1 real places, the corresponding archimedean-and-pole constant is 1 - r1/8. This gives the smooth constants 7/8 for zeta and 1/8 for beta. In the Gaussian field factorization zeta_Q(i)(s) = zeta(s) beta(s), the gamma-factor contributions cancel while the zeta pole-removing contribution remains, giving constant 1. The paper does not claim a new zero-counting estimate. Its contribution is a normalization: it records the additive constants exactly, under one convention, and separates the parts coming from gamma parity and pole removal. Although these constants are usually absorbed into the error term in zero-counting estimates, they must be fixed before finite-interval residuals can be compared. The same normalization fixes the additive baseline used when finite-interval residuals are compared across these functions. The accompanying source and reproducibility bundle includes the LaTeX source, bibliography, numerical spot-check generator, bundled LMFDB reference snapshot, generated table data, verification manifest, SHA-256 checksum manifest, and licence files. Detached GPG signatures for the PDF and source/reproducibility ZIP are uploaded separately. Licensing details for the mixed-content release are recorded in LICENSE-MANUSCRIPT, anc/LICENSE, and anc/LICENSE-DATA.