Exact Complete Relative-Entropy Contraction of Finite Classical Channels

We determine the complete equal-reference Umegaki relative-entropy contraction coefficient of every finite classical channel. For a column-stochastic matrix K with n ≥ 2 inputs, let J consist of the output rows positive in every column. The coefficient is η_c(K) = 1 − min_{p ∈ Δ_n} ∑{y ∈ J} (∑{i=1}^n p_i/K_{yi})⁻¹. The upper bound follows from a dimension-independent residual inequality and the integral representation of the Bogoliubov–Kubo–Mori metric. A weighted simplex construction gives matching faithful state pairs with equal classical and reference marginals; reference dimension n suffices for the supremum. The convergence is pointwise in the channel and weight parameters, with input relative entropy tending to zero. The construction gives no capacity or fixed-error operational guarantee. The complementary loss is the exponential of the negative left order-two Rényi radius of the channel columns. It therefore obeys an exact product law, giving η_c(K ⊗ L) = 1 − (1 − η_c(K))(1 − η_c(L)) even when the optimizing hypotheses may be correlated across uses. We also prove the sharp comparison η_D(K) ≤ η_c(K) ≤ min{1, (n − 1)η_D(K)} with ordinary classical contraction. Exact rational bounds and finite state-pair certificates accompany the analytic proofs. An accompanying three-view visualization guides the reader through the discovery route, proof dependencies, and provenance.