ABSOLUTELY PERIODIC BILLIARD ORBITS OF ARBITRARILY HIGH ORDER

The paper proves the density statement inside D_∞ (unit-length, strictly convex C^∞ planar domains) by constructing absolutely periodic billiard orbits of any prescribed finite order n; this is stated as Theorem 1 and supported by a detailed multi-step proof (tower of heteroclinic tangencies, generic unfoldings, and controlled curvature perturbations), with technical appendices that address key obstacles such as injectivity and near-integrable dynamics near the boundary . The candidate’s solution merely cites precisely this theorem (from the very paper under review) as a black box, then adds a normalization-by-scaling remark. That is circular as a solution to the posed problem. It also glosses over a subtlety: the asserted global linear conjugacy H(s,ϕ)=(λs,ϕ) on T×[0,π] is not a diffeomorphism on the torus unless λ is an integer; only a local coordinate change is justified. The paper’s argument remains consistent and complete at the stated level of rigor, whereas the model provides no independent proof beyond quoting the target result and contains a minor technical inaccuracy about conjugacy. Hence: Paper correct, model wrong.