Interior smoothness of carrying simplices for competitive population maps

A carrying simplex reduces the nonzero long-term dynamics of a competitive population map to an invariant hypersurface. We prove that its positive interior is a $C^1$ embedded manifold under the standard $C^1$ per-capita competition and spectral-radius assumptions. Equivalently, its radial function is $C^1$ in the interior of the probability simplex. The proof uses logarithmic coordinates. Although inverse derivatives lose strict positivity at extinction faces, their weighted columns give a uniform contraction on a fixed cone of positive vectors. The resulting continuous covector annihilates every contingent direction of the logarithmic simplex and forces its Lipschitz graph to be $C^1$. No boundary smoothness, convexity or permanence assumption is imposed.