Dynamics of Non-polar Solutions to the Discrete Painlevé I Equation

The paper rigorously establishes that every Lew–Quarles (positive) orbit of the discrete Painlevé I map converges to P∞ in the (s,f,u) coordinates, i.e., (s,f,u)→(2,2,0) as n→∞, via an adaptation of the original Lew–Quarles contraction mapping and a ratio-limit argument in Appendix B; this is stated as Theorem 2 and proved by showing ξn∼sqrt(n/(3Nr)) and then sm→2, fm→2, um→0 . The candidate solution reaches the same conclusion but its core lower-bound induction x_n ≥ a√n is incorrect: due to the sign structure of the recurrence, replacing x_n and x_{n−1} by their lower bounds produces an upper bound on x_{n+1}, not a lower bound. This flaw vitiates the claimed uniform bound on L_n and the subsequent compactness/limit-point analysis. The paper’s change of variables (s,f,u) is exactly as used by the model and its inverse is given explicitly (equations (11)–(12) in the paper) , but the model’s reliance on a faulty growth lemma leaves its proof incomplete, whereas the paper’s proof is complete in the stated scope.