A concise, general proof of the Multiplicative Ergodic Theorem

The paper’s main theorem (Theorem 3) precisely asserts existence of an equivariant splitting X = (⊕_{i≤l} E_i(ω)) ⊕ V_{l+1}(ω) with Lyapunov exponents defined via Bernstein numbers, identifies the slow spaces, proves temperedness of the projections, and characterizes when the splitting is nontrivial; the statement and the proof structure are explicit in the PDF (see Theorem 3 and adjoining discussion: ; definitions and constants: ; construction of the top fast space and its properties: ; temperedness and induction for the full filtration/splitting: , ; equivalence of growth statistics in the appendix: ). The candidate solution mirrors much of the correct framework (Kingman for µ_k, plateau multiplicities, slow-filtration definition, decomposition criteria) but contains a critical flaw in the temperedness argument: it bounds the projection norm’s exponential rate above by λ_i − λ_{i−1} < 0 and then concludes the limit must be 0 because the projection norm is occasionally ≥ 1. Since any nonzero projection has norm ≥ 1 for all ω, the derived negative upper bound contradicts nonnegativity of 1/n log∥Π_{σ^n ω}∥, and the conclusion does not follow. The paper, by contrast, proves temperedness via an inductive projection comparison that yields 0 ≤ (1/n)log∥Π_{l+1,σ^n ω}∥ ≤ (1/n)log∥Π_{l,σ^n ω}∥ + (1/n)log∥Π'_{σ^n ω}∥ → 0, avoiding this inconsistency (). In addition, the model asserts an identity b_k(T) = c_k(T^*), which is not established in the paper and is not needed; the paper instead proves two-sided comparability between Bernstein and Gelfand numbers up to constants, sufficient for 1/n log-limits (). The model also outsources the core splitting to an external semi-invertible MET reference rather than establishing it from the paper’s tools (balanced Kingman plus contraction of fast regions). Hence, the paper is correct and complete for its aims, while the model’s solution has a material error and gaps in justification.