Fixed-point equalization in diagonal expectation propagation: Scalar decoupling and Bayes-MMSE optimality

We analyze diagonal Expectation Propagation (EP) at fixed points for linear inverse problems with separable priors and right-orthogonally invariant (ROI) sensing. Starting from exact finite-dimensional extrinsic identities, a simple leave-one-out (LOO) concentration establishes equalization of the likelihood-site precisions. A Haar-based central limit theorem then shows that extrinsic residuals are Gaussian and white, yielding an effective scalar AWGN channel. With Bayes denoisers, the prior site equalizes and a single-scalar consistency links the effective noise level to the Bayes MMSE, so that the fixed-point mean-squared error attains the Bayes-optimal value. Our analysis is deliberately simple: it requires asymptotics only for the LOO step and avoids site uniformity and free-probability averaging. We characterize fixed points rather than convergence; algorithmic issues such as damping or projection are left for future work.