Statistical-physics calculations in machine learning and theoretical neuroscience often involve lengthy derivations that obscure physical interpretation. We present concise, non-replica derivations of key results and highlight their underlying similarities. Using a cavity approach, we analyze high-dimensional learning problems: perceptron classification of points and manifolds, and kernel ridge regression. These problems share a common structure--a bipartite system of interacting feature and datum variables--enabling a unified analysis. For perceptron-capacity problems, we identify a symmetry that allows derivation of correct capacities through a na\"ive method.

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