Split Prime Divisors in Cyclotomic Norm Layers: A Sharp Answer to Mossinghoff–Pinner Question 4.9

This preprint establishes a sharp bound for the number of cyclotomic norm layers that can fail to contain a rational prime divisor q congruent to 1 modulo p^i. Let p be an odd prime and let N_i(F) denote the norm of F evaluated at a primitive p^i-th root of unity. Under the conditions v_p(F(1))=k≥1 and v_p(N_i(F))=1 for 1≤i≤t, at most k layers are bad for p≥5, and also for p=3 when k≥2. In the remaining case p=3, k=1, at most one of the eligible layers 2,…,t is bad. The bounds are sharp in every counted range. With k=j−t, the theorem gives an affirmative answer to the contextually intended p^i formulation of both parts of Mossinghoff–Pinner Question 4.9. The proof combines faithful-character descent for p-adic logarithms with an integral primitive-trace projector, shifted cyclotomic divisibility, and an exact boundary-valuation contradiction.