The Pre-Gauge Gamma Bridge: Kummer Structure and the Critical-Line Gamma Product

Research Note 33 in the "Geometry of the Critical Line" programme. This note resolves the archimedean mismatch that arose from the post-gauge Bessel/Whittaker reduction (Papers 41–43, RN15). In the gauge-fixed frame, the endpoint Bessel order has real part approximately 1, obstructing comparison with the Riemann archimedean L-factor. This note shows that the obstruction is a gauge artifact: in the pre-gauge frame, the Frobenius exponents have real part exactly 1/2, and the endpoint equation reduces to a confluent hypergeometric (Kummer) equation whose connection coefficients are Gamma functions of the form Γ(1/2 + it) with t real — structurally identical to the archimedean L-factor on the critical line. The Gamma duplication formula decomposes each factor into both parity sectors of the Connes archimedean space, providing the first direct structural link between the SCT endpoint geometry and the Riemann completion factors.