PREPERIODIC POINTS WITH LOCAL RATIONALITY CONDITIONS IN THE QUADRATIC UNICRITICAL FAMILY

The paper establishes two precise trichotomies for totally real and totally p-adic preperiodic points of f_c(x)=x^2+c over Q, stated as Theorem 1.1 and Theorem 1.3, with clear proofs via the archimedean/nonarchimedean shapes of filled Julia sets and an adelic Fekete-type finiteness argument (Baker–Hsia) for the finiteness cases. See Theorem 1.1 and its proof outline (finite/nonempty regime handled in §3 via Proposition 3.1) and the archimedean shape result Proposition 2.1 for c>1/4 and c≤−2, and Theorem 1.3 together with Theorem 5.1 (Benedetto–Briend–Perdry) for the p-adic trichotomy and its consequences for total p-adicity and emptiness/infinitude . The candidate solution reaches the same statements but its proofs contain critical flaws: (i) the equidistribution argument at the good-reduction p-adic place uses Urysohn’s lemma to separate P^1(Q_p) from the Gauss point, but P^1(Q_p) is dense in the Berkovich projective line, so such a continuous separator cannot exist; equidistribution alone does not yield the claimed contradiction. The paper’s capacity/Fekete argument covers this case correctly (Theorem 3.2(b) analog and Theorem 5.2(a)) . (ii) In the |c|_p>1, −c non-square case, the model relies on a false nonarchimedean claim (“if |a|=|b| and a≠−b then |a+b|=|a|”); the paper instead relies on a correct dynamical classification (BBP) . The model also omits an explicit Northcott-type degree-growth justification needed to apply equidistribution at ∞ in the totally real finiteness step, while the paper’s approach avoids this issue entirely via transfinite diameter . Hence, the paper’s results and proofs are correct, while the model’s proof contains a substantive error in the p-adic good-reduction case and smaller gaps elsewhere.