THE JOINT TRANSLATION SPECTRUM AND MANHATTAN MANIFOLDS

The paper proves that Int(Jψ(D)) = {−∇θD/ψ(v) : v ∈ R^n} and that −∇θD/ψ is a homeomorphism onto Int(Jψ(D)) under independence, via Manhattan manifolds, large deviations, and geodesic currents; see Theorem 1.12 and its development: θ is C1 (Proposition 5.7), a large deviation principle pins down Jψ(D) as the gradient image (Proposition 5.11), and a boundary-currents argument yields surjectivity/homeomorphism (Proposition 6.11) . Foundationally, the joint spectrum exists, is convex/compact, and has nonempty interior under independence (Theorem 1.4) . The candidate solution reaches the same conclusions using a different route (SFT coding, pressure, rotation sets, and Legendre duality). Its outline is mathematically sound in classical thermodynamic-formalism settings, but it implicitly assumes a mixing SFT coding and Hölder/Bowen properties for the induced observables on the coding, and the identification Jψ(D) = Rot_g(F); these hypotheses are not stated or justified in full generality for all ψi, ψ ∈ HΓ. Hence, the paper is correct as written, and the model’s proof is correct in principle but requires additional hypotheses and technical justification.