Exact Archimedean Flicker--Rallis Periods on Cohomological Minimal $K$-Types

For the symmetric pair $({\mathrm{G}\mathrm{L}}_n(\mathbb{C}),{\mathrm{G}\mathrm{L}}_n(\mathbb{R}))$, we compute the exact Flicker–Rallis period of a distinguished vector in a cohomological minimal $U(n)$-type, normalized by the Gel’fand–Tsetlin pairing of Hara–Miyazaki–Namikawa. Our main tool is a weight-independent identity between the Whittaker period and a compact $O(n)$-average. We determine the proportionality constant using the archimedean newform formula of Humphries–Jo, and evaluate the compact functional on the normalized distinguished vector. This yields an explicit formula for the Flicker–Rallis period for every cohomological weight.