Cyclotomic Specialization of Unicritical Multiplier Curves and the Morton–Vivaldi Conjecture in Period 4

For the unicritical family f_c(z) = z^d + c, we introduce a reduced multiplier on its multiplier curves. For cycle period m, with gamma = gcd(m, d − 1), the usual multiplier is the gamma-th power of the reduced one. We prove that every power pullback of the reduced map is geometrically irreducible and that no larger power can be removed from the usual multiplier. Torsion separation and cyclotomic Hilbert irreducibility then give, outside finitely many orders, complete factorization formulas over Q, Q^ab, and fixed number fields. Thus each quadratic period row of the Morton–Vivaldi conjecture has only finitely many exceptions. We determine all exceptions in period 4: every period-4 delta factor is irreducible over Q. More strongly, for every root of unity zeta ≠ 1, the associated sextic over Q(zeta) has Galois group containing A_6, remains irreducible over the maximal solvable extension of Q(zeta), and has splitting group A_6 there. This yields exact factorizations after solvable base change and the maximal solvable subfields of the associated parameter fields. The generic splitting field is a regular S_6-extension, and all but finitely many torsion specializations retain group S_6 over Q^ab.