A Dynamical Analogue of the Criterion of Néron-Ogg-Shafarevich

The paper’s main theorem (Theorem 3.1) states precisely parts (i)–(iii) that the candidate solution establishes: if a branch is not eventually residually stepwise simple, then the residual branch is post‑critical and periodic, the induced branch representation for P^m is eventually totally ramified with degree equal to the Weierstrass degree of P^m(x+a)−a, and consequently the original branch representation is infinitely ramified. This appears verbatim in the abstract and Section 3 of the paper, with the proof reducing to the residually constant case via Proposition 2.10/Lemma 2.7, then applying a power‑series/Weierstrass argument (Proposition 2.5) to obtain eventual total ramification of degree e, and finally deducing infinite ramification from iteration . The model’s solution follows the same structure and key ideas: (1) periodicity via recurrent residual critical points (height 0 case) then untwisting by Frobenius (general height), aligning with Lemma 2.7 and Proposition 2.10 ; (2) expansion at a periodic residual critical point and Weierstrass preparation to read off the Weierstrass degree and hence the asymptotic ramification; this matches the paper’s use of Proposition 2.5 ; and (3) infinite ramification by seeing nontrivial ramification at every mth step. Minor gaps in the model’s write‑up (e.g., implicitly assuming Frobenius commutes with the residual map without first passing to a suitable iterate as in Proposition 2.10) are easy to fix with the paper’s standard device of iterating until Φ^{hk} fixes coefficients. Overall, both arguments are correct and essentially the same.