On the Distribution of the Cyclotomic Norm Form n⁴⁷ − (n−1)⁴⁷ in Arithmetic Progressions

We study the polynomial Q(n) = n⁴⁷ − (n−1)⁴⁷, a degree-46 norm form from the cyclotomic field ℚ(ζ₄₇). We prove that the root count ω(p) = 0 for all primes p ≢ 1 (mod 47) and p = 47, while ω(p) = 46 for p ≡ 1 (mod 47). This complete local obstruction defines a sparse set of effective moduli 𝒬_eff — integers whose prime factors all satisfy p ≡ 1 (mod 47) — with counting function ≍ D(log D)−45/46. Exact computation confirms that 99.8% of moduli q ≤ 10⁶ are null (contributing zero to the Bombieri–Vinogradov error sum). We prove a null–sparse decomposition theorem: the BV error sum splits into a null part (where the main term vanishes identically) and a sparse part (supported on 𝒬_eff). A Cauchy–Schwarz argument with a global Barban–Davenport–Halberstam hypothesis yields an exponent of +1/92, which barely fails to reach θ = 1/2. Under a restricted variance hypothesis, a double sparsity factor produces the exponent −11/23, achieving θ = 1/2 with a power saving of (log x)−11/23. The repository includes the full paper (LaTeX source and compiled PDF), three CSV data files (local root structure for 20 splitting primes, effective moduli counts up to D = 10⁶, Cauchy–Schwarz exponent comparison table), and three Python verification scripts. No unconditional improvement beyond the classical θ = 1/2 is claimed.