Proximally irreducible affine extensions: composition and maximal covers

We prove that the composition of proximally irreducible affine extensions is proximally irreducible, answering a question of Bal for arbitrary topological groups and compact Hausdorff affine flows. The proof combines localization, Baire category, and compactness for finite convex witnesses, without requiring closed extreme-point sets. Composition, factorization, inverse limits, and uniform density bounds yield maximal proximally irreducible covers, unique up to a unique isomorphism over the base and characterized by a universal property. Extensions of a fixed base form a complete lattice, realized by closed invariant spaces of affine functions on its maximal cover. Passing to probability flows recovers maximal proximal covers of minimal flows and gives canonical refinements, pullbacks, and functorial lifts. For the trivial group, Gleason covers describe maximal domains and classify polytope extension lattices. For separable groups, extensions of metrizable bases admit metrizable approximations that detect nontriviality. Examples delimit the affine pullback and metrizability statements.

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