Construction and applications of proximal maps for typical cocycles

The paper proves the uniform periodic approximation theorem (Theorem A) by constructing periodic points whose exterior-power products are simultaneously proximal, then comparing log norms and log spectral radii via a bounded distortion/shadowing argument; the proof is coherent and complete. The candidate solution proposes an alternative route via simultaneous quasi-multiplicativity and specification, but it contains a critical gap: it does not rigorously justify the lower bound linking the log spectral radius of the periodic product J_t(A^{n_q}(q)) to the subadditive singular-value sum S_t(A^n(x)) using powers of the periodic word without additional connectors. It also asserts, without proof, that a fixed holonomy–pinching loop W yields wedge-power maps with simple dominant eigenvalues. These issues make the model’s proof outline incomplete.