Empirical Verification of the Bateman-Horn Conjecture for the Degree-46 Polynomial Q₄₇(n) = n⁴⁷ - (n-1)⁴⁷

We present a large-scale computational verification of the Bateman-Horn conjecture for the degree-46 polynomial Q₄₇ (n) = n⁴⁷ - (n-1) ⁴⁷ in the quadruplet configuration. By executing a systematic sieve over the range n ∈ 2. 3 × 10⁷, 8 × 10¹¹, we discovered 2, 359 prime quadruplets. Furthermore, using a targeted Chinese Remainder Theorem (CRT) search, we found eleven 1000-digit prime quadruplets at n ≈ 4. 8 × 10²¹, all 44 constituent primes of which have been rigorously certified via Elliptic Curve Primality Proving (ECPP). We evaluate the Bateman-Horn structural constant CBH ≈ 6514. 2 utilizing an Euler product over 5. 76 × 10⁶ primes, incorporating an Abel summation tail correction to handle conditional convergence. A standard χ² goodness-of-fit test comparing observed versus predicted counts across 12 data sectors yields χ² = 10. 56 on 11 degrees of freedom (p ≈ 0. 48), demonstrating excellent agreement with the asymptotic prediction.