Prime Values of a Cyclotomic Norm Polynomial and a Conjectural Bounded Gap Phenomenon

We study the polynomial Q(n) = n⁴⁷ − (n−1)⁴⁷, which arises as a norm form from the cyclotomic field ℚ(ζ₄₇). We prove a complete local obstruction property: for any odd prime p, the congruence Q(n) ≡ 0 (mod p) has solutions if and only if p ≡ 1 (mod 47), and 47 never divides Q(n). This yields an enhanced Bateman–Horn constant C_Q ≈ 8.64, verified against ~10⁶ primes with relative error below 0.1%. We establish the algebraic prerequisites for sieve methods (shifted irreducibility, absence of fixed common divisors) and prove the Titan–BV Decomposition Theorem: ~97.8% of moduli contribute exactly zero error to the Bombieri–Vinogradov sum, reducing the analytic problem to a sparse residual over cyclotomic moduli. Direct computation of exponential sums over all 20 bad primes p ≤ 6299 confirms square-root cancellation with an 80% safety margin below the Hasse–Weil bound. The repository includes the full paper (LaTeX source and compiled PDF), seven CSV data files covering all numerical tables in the paper, and four verification scripts (Python and SageMath) for independent reproducibility. No unconditional bounded gap result is claimed.