Entropy formula of folding type for C1+α maps

The paper establishes both sufficiency and necessity (under (H)”) of the folding entropy formula for all C^{1+α} maps, with complete proofs built on Pesin theory for degenerate maps, stable-partition disintegration, and an inverse-limit argument. The candidate solution reaches the same conclusions but relies on an unjustified identity h_μ(f) = F_μ(f) − ∫ log J_μ(f|ξ) dμ. The paper does not assert this identity; instead, it shows F_μ(f) = H_μ(ε|f^{-1}ε) = H_μ(η|f^{-1}η) + ∫ Σ_{λ_i<0} λ_i via a carefully derived leafwise formula and then uses H_μ(η|f^{-1}η) ≤ h_μ(f) to obtain equality by sandwiching with Ruelle’s folding inequality. The model also glosses over the technical construction of stable manifolds and subordinate partitions in the degenerate setting. Hence, the model’s proof is incomplete/incorrect in a key step even though its final conclusions match the paper.