We study binary discrimination of idempotent quantum channels. When the two channels share a common full-rank invariant state, we show that a simple image inclusion condition completely determines the asymptotic behavior: when it holds, a broad family of channel divergences collapse to a closed-form, single-letter expression, regularization is unnecessary, and all error exponents (Stein/Chernoff/Hoeffding/strong-converse) are explicitly computable with no adaptive advantage. Crucially, this yields the strong converse property for this channel family. When the inclusion fails, asymmetric exponents become infinite, implying perfect asymptotic discrimination. We apply the results to GNS-symmetric channels, showing discrimination rates for large number of self-iterations converge exponentially fast to those of the corresponding idempotent peripheral projections. Without a common invariant state, we show that the same image inclusion still guarantees additivity of the sandwiched Rényi cb-divergence for every $α>1$, which suffices to establish the strong converse property and single-letter expressions for the Stein exponents.

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