Natural extensions and entropy of α-continued fraction expansions with odd partial quotients

The paper explicitly constructs the natural extension (Ω_α, T_α) for odd α-continued fractions across the range √13−1 over 6 ≤ α ≤ G, proves it carries the invariant density (3 log G)^{-1}(1+xy)^{-2}, shows bijectivity mod 0 and metric isomorphism among all parameters in the range, and concludes the entropy is π^2/(9 log G) for all such α. The candidate solution reproduces the same construction (via the same Möbius branch formulas, the same auxiliary maps M_α and A, and the same union–difference description of Ω_α), the same invariant density calculation, the same natural extension/factor properties, and the same parameter-independent entropy (computed at α=g using elementary dilogarithm identities). The only substantive difference is stylistic: the model frames the isomorphism via a common bi-infinite coding, whereas the paper establishes it geometrically via singularizations/insertions and explicit planar domains. No contradictions were found.