Innovative Dynamics: Utilizing Perelman’s Entropy and Ricci Flow for Settler Position Models on Manifolds

The paper asserts W(g,f,τ) ∼ O(e^{κ1 τ}) and even states that the entropy increases exponentially, but it replaces the spatial gradient by a time derivative (|∇f|^2 = (df/dτ)^2) and then integrates over R^3 with f depending only on τ, which makes the spherical integral ∫_0^∞ r^2(⋯)e^{-f(τ)}dr diverge unless additional spatial localization is imposed; moreover, Big-O is misinterpreted as an increasing trend. These issues are visible where the paper defines |∇f|^2 via df/dτ and converts to the radial integral over R^3, then concludes O(e^{κ1 τ}) and “entropy increases exponentially” for large τ . The model (candidate) supplies a rigorous upper bound W ≤ C e^{κ1 τ} by exploiting the super-exponential damping e^{-f(τ)} and explicitly adds a finite-support/normalized radial density assumption, which fixes the divergence, and clarifies sign conditions on c1 and κ1. However, it inherits the paper’s nonstandard identification |∇f|^2 = (df/dτ)^2 and does not reintroduce a proper spatial profile for f. Net: the paper’s argument is mathematically flawed/incomplete, and the model’s bound is correct only under extra assumptions and the same gradient simplification, so both are incomplete.