Unicritical Polynomials over abc-fields: From Uniform Boundedness to Dynamical Galois Groups

The paper proves the three statements the model sketches, and more. In Theorem 1.2 it establishes, for abc-fields and all sufficiently large d with p ∤ d, that (1) there are no K-rational cycles of period > 3 for nonzero (and non-constant, in function fields) c; (2) for large h(c), PrePer(x^d+c,K) equals {ζy: ζ in μ_{K,d}} with a unique fixed point y and c=y−y^d; and (3) for small h(c), all preperiodic points have height 0; moreover, in function fields one can take C1=0 and D1=max{8, 2g_K+6} . The proof machinery includes precise height inequalities (e.g., Corollary 3.4) and a reduction to Fermat–Catalan-type equations, yielding the large-height classification and the function-field explicit constants . Beyond this, the paper classifies preperiodic skeleta up to isomorphism (thirteen possibilities) for all sufficiently large d (Theorem 4.3), a result not present in the model’s outline . By contrast, the model’s "likely open as of cutoff" assertion is incorrect, and its argument relies on unproven coprimeness claims and unnecessary restrictions (e.g., d ≢ 1 mod p).