Finite index theorems for iterated Galois groups of preperiodic points for unicritical polynomials

The uploaded paper proves that for f(x)=x^q+c with q a prime power, 0 not preperiodic, and β strictly preperiodic, the index [G∞:G∞(β)] is finite (Theorem 1.2) and reduces this to certifying that, for all large n, Gal(Kn(β)/Kn−1(β)) attains the full Kummer kernel (Cq)^{q^{n−1}} (Proposition 2.1) by producing primes satisfying Conditions R/U and a multi-branch independence criterion (Propositions 4.7 and 4.9, combined with Proposition 5.1 and Theorem 5.3) . By contrast, the candidate solution makes a crucial incorrect reduction: it replaces β by a periodic α=f^m(β) and claims the towers agree from level m onward, which is false in general, and it does not supply the paper’s precise Conditions R/U or eventual stability/irreducibility inputs. It also attributes the key prime construction to a different source (squarefree portrait primes) than the paper’s actual Proposition 3.1 mechanism. Thus the paper’s argument is correct and complete for its claims, while the model’s proof outline has a fundamental flaw and missing hypotheses.