Feature Learning in Wide Neural Networks under μP: Identifiability and Sparse-Dictionary Decomposition of the Mean-Field Limit

We establish four structural results for feature learning in wide two-layer neural networks under the Maximal Update Parametrization (μP). First, we prove global existence and uniqueness of the mean-field limit of noisy gradient descent under μP, identifying the maximal admissible weight w^* on the moment sequence of the initialization as the reciprocal parameter-moment-growth boundary, and hence the largest weighted moment class propagated by the flow. The finite-particle approximation has uniform-in-time squared-Wasserstein rate O(N-1). Second, we characterize identifiability of the mean-field limit: two admissible parameter measures induce the same network function in L2 exactly when their active components agree modulo the finite-rank realization symmetry of the architecture. The orbit depth D^*orb is separated from the moment-variety depth D^*var. Third, under the Barron-Hermite target condition the active support of the long-time limit measure admits a sparse-dictionary decomposition: it is supported on at most S^* atoms modulo finite-rank realization symmetry, with S^* bounded by an explicit coefficient-threshold number. Fourth, we derive the total feature-learning-error decomposition into statistical, optimization, propagation-of-chaos, and sparse-residual components, with a target-dependent Hermite/Barron tail replacing any initialization-only residual. The four results are tied together by an architectural identity: the triple (w^*, D^*orb, S^*) -- the maximal admissible weight, the orbit identifiability depth, and the sparse-dictionary depth at which the target is realizable -- is the natural learning cell of the architecture-data pair (σ, ρ). The proofs are self-contained except for standard results from μP and mean-field Langevin theory.