Global Pre-Gauge Composition and the Critical-Line Gamma Product

Research Note 34 in the "Geometry of the Critical Line" programme. RN33 showed that the SCT endpoint geometry produces Gamma factors of the form Γ(1/2 + it) with t real. But an Evans function is a global object — endpoint matching alone does not determine whether those Gamma factors survive transport through the nonsingular interior. This note separates the two roles: the singular endpoints contribute the Gamma prefactor, while the smooth interior contributes a regular holomorphic multiplier R(λ,m). The holomorphicity of R follows from standard ODE theory (smooth coefficients on a compact interval). The quantitative behaviour of R — whether it is slowly varying, oscillatory, or has zeros interacting with the Gamma prefactor — is identified as the main remaining open problem. This is the minimal global statement needed before any comparison with an archimedean L-factor can proceed.