p-ADIC INTERPOLATION OF ORBITS UNDER RATIONAL MAPS

The paper’s Theorem 1.1 states precisely the claimed interpolation along arithmetic progressions and proves it by (i) constructing a Zariski-dense set of primes via an S-unit argument (Lemma 2.2), (ii) finding an iterate/point satisfying h(c) ≡ c (mod π^2) and h′(c) ≡ 1 (mod π), and then (iii) applying Poonen’s p-adic interpolation theorem after a careful normalization (Lemma 2.3 and Proposition 2.4) to produce g_i with h^{a n + i}(c) = g_i(n) for large n . The model’s proof tries to force derivative ≡ 1 mod v at a Teichmüller lift and then ‘scale coordinates’ to meet Poonen’s hypotheses; however, scaling that improves higher-order coefficients simultaneously worsens the constant term, and without ensuring h(c) ≡ c (mod π^2) (or an actual fixed lift) one does not get f(x) ≡ x (mod π^c) for all coefficients as required by Poonen. This gap is exactly what the paper’s S-unit/Zariski-density lemma repairs. Hence the paper is correct, while the model’s argument is incomplete/incorrect in the key analytic step .