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…