Rigorous State Evolution for General-Core Asymmetric Multilinear AMP

State evolution (SE) provides an asymptotically exact finite-dimensional characterization of approximate message passing (AMP) dynamics in many high-dimensional inference problems. For asymmetric multilinear AMP, however, reusing the same Gaussian tensor across mode contractions and iterations creates dependencies that complicate both the Onsager correction and rigorous SE analysis. Existing tensor-AMP results do not directly cover general-core rectangular models with empirical covariance inputs and Onsager reaction matrices. We derive core-induced matrix-valued maps that govern the signal contribution and Gaussian covariance and whose derivatives yield an exact finite-system factorization of the Onsager reaction. For fully observed Gaussian models of order at least three with a fixed known core and fixed, mode-dependent channel dimensions, we prove fixed-horizon SE for the joint empirical law of planted rows, factor-iterate histories, and effective-field histories within each mode. The theorem allows initialization correlated with the planted factors but independent of the Gaussian disorder. The closed SE recursion tracks signal and iterate overlaps separately and does not require Bayes matching. The proof combines a symmetric multi-species embedding and scalar-query serialization with Gaussian conditioning supported by quantitative bounds on the stacked-response covariance in an auxiliary regularized model. The joint law yields limits for representation statistics and simultaneous convergence of order-two pseudo-Lipschitz losses over fixed finite families of admissible recursions and independently fitted, frozen terminal readouts. Numerical experiments assess SE predictions for non-diagonal cores and covariance-dependent updates with unequal channel dimensions, and illustrate estimation gains from SE-calibrated terminal history readouts.