Non-Birkhoff periodic orbits in symmetric billiards

The paper proves Theorem 10.1 (and its formulation as Theorem 1.1) via a precise Hessian computation at the D_n-symmetric Birkhoff orbit (Lemma 8.4), an eigenvector choice aligned with the dihedral symmetry, and a gradient-flow/monotonicity argument (Sturmian lemma) that converges to an H-symmetric non-Birkhoff orbit of minimal period p = sn when gcd(s,N) = 1. The inequality κL < 2 sin(mπ/n) cos^2(Nπ/p) is exactly the positivity condition for the relevant Hessian eigenvalue and underpins the construction (Theorem 10.1; proof steps around (33)–(37), and the period argument via lcm(n, K)) . By contrast, the model’s proposal hinges on an unsupported sign-flip claim for the two D_n-symmetric seeds (c_- = -c_+) and a symmetric mountain-pass on an asserted 2D H-fixed submanifold. The paper neither needs nor justifies such a sign flip; the eigenvalue used is the same sign for the chosen seed once κL is below the stated threshold (see the explicit λ(N) formula), and the existence proof proceeds by gradient flow rather than a mountain-pass. The model also gives an unnecessary characterization of H-symmetric Birkhoff orbits; the paper uses the simpler fact that p and q are not coprime, hence the obtained orbit cannot be Birkhoff (Proposition 3.5) . Overall, the paper’s argument is complete and correct; the model contains critical gaps and inaccuracies.