Cartan calculus in tangent categories

What the math gives to ML

The paper supplies an abstract algebra of tangent vectors in which vector fields act as derivations on scalar functions and their Lie bracket obeys the Lie-Rinehart Leibniz rule. The transferable asset is not the categorical generality itself, but a principled way to parameterize state-dependent vector-field modules while preserving differentiation, scaling, and commutator identities. A neural implementation can use learned tangent/JVP operators and vector-field generators, then enforce these identities as architectural constraints or consistency losses. This is most relevant to neural dynamical systems, latent-state models, and models that compose multiple learned infinitesimal transformations.