Totally Real Algebraic Numbers in Generalized Mandelbrot Set

The paper proves Theorem 1.4 exactly as stated in the candidate solution: for α ∈ Q, Prep(α) ∩ Q_tr is finite and non-empty when |α| < 2, and infinite when |α| ≥ 2 (see Theorem 1.4 and its proof, together with Lemma 4.1 and Theorem 1.2) . Both arguments rely on: (i) the local–global principle in parameter space (Lemma 4.1), (ii) capacity one for all local Mandelbrot sets (Proposition 2.1), (iii) Baker–DeMarco’s infinitude of preperiodic parameters, and (iv) the fact that M(α) ⊂ R for |α| ≥ 2 . The model’s proof mirrors the paper’s Fekete–Szegő approach for |α|<2 and the real-slice argument for |α|≥2. Minor issues: the model’s “capacity < 1 ⇔ interval length < 4” claim is not needed and false in general, and the paper’s invocation of monotonicity to assert strict capacity inequality at ∞ is a bit terse; however, the paper supplies an alternative equidistribution argument that closes any gap (Remark 4.2) .