Singular Dynamics for Discrete Weak K.A.M. Solutions of Exact Twist Maps

The paper proves, under Hypotheses 1–3, that the forward selector Σ^+ is single‑valued, non‑decreasing, Lipschitz, propagates singularities, and has a rotation number equal to α_1(c), and it further shows that iterates from a singular point stay singular with asymptotic slope α_1(c) (Theorems 1.1–1.2). These statements and their proofs are explicit and coherent: monotonicity via the non‑crossing lemma (Lemma 3.1), single‑valuedness via a backward–forward calibration argument (Lemma 3.2), global Lipschitz continuity via Lipschitzness on a calibrated set plus a covering argument (Proposition 3.5), propagation of singularities via a precise characterization of preimages (Proposition 3.6 and Corollary 3.8), and rotation number existence/identification via standard degree‑one monotone map theory and Aubry calibrations (Lemma 3.9, Proposition 3.10, Theorem 1.1(iv)) . By contrast, the candidate solution contains crucial gaps and over‑claims: (i) its uniqueness proof for the maximizer asserts equality of envelopes on an interval without justification; the paper’s proof avoids this pitfall by using the sub/super‑solution interplay (Lemma 2.5) and semiconcavity to rule out multiple maximizers (Lemma 3.2) ; (ii) its global Lipschitz estimate relies on a uniform mixed‑increment bound that does not follow from Hypothesis 2 alone (C^1 ferromagnetic) and tacitly uses C^2/∂^2_{xy}S≤−ε; the paper instead assumes local Lipschitz regularity of the standard map and builds the estimate rigorously (Proposition 3.5) ; (iii) its propagation-of‑singularities argument invokes a forward–backward conjugacy u=T^+u−α(c) that is not established in this discrete setting, whereas the paper proves propagation without that assumption (Proposition 3.6, Corollary 3.8) . Therefore, while the model states the correct conclusions, its proofs are invalid under the stated hypotheses, whereas the paper’s arguments are correct and complete.