Large-system fixed-point law and deterministic closure for sparse bayesian learning

Sparse Bayesian Learning is widely used for high-dimensional inverse problems, yet fixed-point behavior in underdetermined systems is poorly understood. We give a distributional account of the hyperparameters at convergence under independent Gaussian designs with deterministic heterogeneous signals. Each coordinate follows a spike-and-tail law with a translated noncentral chi-square tail. A quenched selfaveraging principle turns empirical averages into deterministic quantities and yields a compact closure based on three resolvent-derived scalars, predicting activation, error, and detection without a signal prior. With learned noise variance, a simple balance fixes the effective activation threshold at one. Simulations validate the coordinate law and the closure.