The Bateman--Horn Constant as a Compression Factor: Prime Density Enhancement in the Titan Polynomial

The Bateman–Horn conjecture predicts that the density of prime values of an irreducible polynomial f(n) is governed by a singular series constant S(f). For the Titan polynomial Q(n) = n⁴⁷ − (n−1)⁴⁷, of degree d = 46, the Shielding Property (ω_Q(p) = 0 for all primes p < 283) produces an anomalously large constant S(Q) ≈ 8.70. All 60 primes below 283 — including the ramified prime p = 47 — contribute a factor p/(p−1) > 1 to the Euler product, while the sparse splitting primes p ≡ 1 (mod 47) contribute corrective factors (p−46)/(p−1) < 1. We quantify the resulting "Titan Compression Effect": at magnitude x = 10^10000, the expected gap between prime-producing arguments is H* ≈ 121,781 for Q(n), compared to H* ≈ 1,059,196 for a generic degree-46 polynomial — an 8.7× compression. Equivalently, Q(n) behaves like a polynomial of effective degree ≈ 5.3 in terms of prime density. For comparison, the Hardy–Littlewood twin prime constant C₂ ≈ 1.32 is an order of magnitude smaller, though the two constants govern fundamentally different problems. The repository includes the paper (4 pages), three data CSVs (convergence of S(Q) for 9,592 primes up to 10⁵, gap compression table, summary statistics), two PDF vector figures (convergence curve and stress test plot), and two Python scripts for reproducing all computations and figures.