Trajectory convergence from coordinate-wise decrease of general convex energy functions

The paper’s Theorem 1 and its proof (via K-minimal accumulation points, local Taylor bounds, and a closure/density argument) are sound and consistent with the stated assumptions. By contrast, the model’s proof hinges on two unjustified steps: (i) it erroneously deduces that the gradient is constant on the ω-limit set by claiming equality in the convex first-order inequality, and (ii) it asserts V is affine along segments between accumulation points, thereby placing q−p in ker ∇²V(p). These steps do not follow from convexity and lead to an incorrect argument; the model also implicitly assumes compact sublevel sets to conclude convergence when Ω is a singleton.