Maximal Generic Iterated Galois Images Do Not Determine p-Adic Julia Sets

An AIM problem asks whether the arithmetic or geometric Galois group of the generic iterated-preimage tower of a rational map over a p-adic field determines its Julia set. We give a uniform negative answer. For every prime p, the quadratic polynomials z² + 1 and z² − p⁻⁶ over ℚ_p have arithmetic and geometric generic iterated Galois images equal to the full group Aut(T₂). The first map has good reduction: its Berkovich Julia set is the Gauss point and its classical Julia set is empty. The second has a type-I Cantor Julia set; moreover, its two contracting inverse branches are defined over ℚ_p, so the classical Cantor set already lies in P¹(ℚ_p). Thus even the full rooted-tree action and the marked normal pair of arithmetic and geometric images fail to determine the cardinality or topological type of the Julia set. The result combines Pink's maximality theorem with a base-field inverse-branch construction. This note is unrefereed and makes no absolute priority claim. Research status: Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-use disclosure: AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text.