GEODESICS AND DYNAMICAL INFORMATION PROJECTIONS ON THE MANIFOLD OF HÖLDER EQUILIBRIUM PROBABILITIES

The paper proves a local existence–uniqueness theorem for geodesics (Theorem 1.2) on the manifold N of Hölder equilibrium probabilities under a “Fourier-like basis” assumption, using an analytic, PDE/ODE scheme built on finite-dimensional projections and a Picard-type fixed-point argument on spaces of analytic Hölder curves. It explicitly avoids relying on classical Hilbert-manifold geodesic theory because the natural L2 metric is weak relative to the Hölder (Banach) manifold structure and the standard Levi-Civita/Christoffel ODE approach does not directly apply. See the statement of Theorem 1.2 and the Fourier-like hypothesis, their analytic chart/projection Π and bounds (Proposition 2.4), and the Picard theorem and geodesic existence/uniqueness steps (Section 3) . By contrast, the model’s solution assumes a strong, real-analytic Hilbert metric on a Hilbert model space E and then applies the standard Levi-Civita/geodesic ODE well-posedness. This silently replaces the underlying Hölder Banach structure by an L2-based Hilbert structure and presumes the Riesz isomorphisms and Christoffel maps are well-defined and locally Lipschitz in the appropriate (Hilbert) topology—points the paper flags as nontrivial and avoids. Without proving that the L2 metric is strong relative to the chosen manifold topology, the model’s argument does not go through in this setting. Hence: paper correct; model’s proof relies on unestablished assumptions.