Satellite Primes: The Local Prime Landscape around Giant Primes from Q (n) = n⁴7 - (n-1) ⁴7 — A Cramér-Model Validation at 500-Digit Scales

We report the first large-scale empirical study of the local prime landscape surrounding giant primes P = Q (n) = n⁴⁷ - (n-1) ⁴⁷, probing the nearest primes within radius R = 5, 000 of each P. For 2, 107 main-star primes P of 494–521 digits, 9, 012 satellite primes P - k were discovered via probabilistic primality testing. The satellite count per star follows a Poisson distribution with λ = 4. 28 and dispersion index 1. 07. The gap distribution confirms the Cramér random model at 500-digit scales (χ² test, p = 0. 31). All gaps satisfy k ≡ 0 or 2 (mod 6), a forbidden residue lattice arising from Q (n) ≡ 1 (mod 6). This lattice concentrates satellite density onto 1, 667/2, 500 admissible slots at rate ~3/ln (P) each. A confirmation scan at R = 100 across all 2, 992 main-star primes reveals: - 7 twin prime pairs (k = 2): 500-digit primes separated by 2- 7 sexy prime pairs (k = 6): 500-digit primes separated by 6 Both counts match the conditional Hardy–Littlewood expectation E ≈ 7. 2 to within 0. 1σ, providing a precision validation of the Bayesian concentration principle: the fixed residue P ≡ 1 (mod 3) doubles the conditional rate for k ≡ 2 (mod 6) gaps, exactly compensating the smaller unconditional singular series, so that Scond (k=2) = Scond (k=6) = 2. 64. The 3-smooth baseline family (gaps k = 2ᵃ × 3ᵇ) all share the same conditional expectation E = 7. 2, explaining why k = 8 (2 observed) is rarer than k = 2 (7 observed) despite identical theoretical predictions — pure Poisson fluctuation. These results establish that the algebraic origin of giant primes from a high-degree polynomial does not perturb their local prime environment: Cramér universality holds at 500 digits. Package contents: LaTeX source and compiled PDF (14 pages, 4 figures, 8 tables), Python scanning scripts (R=5000 and R=100), complete satellite data (9, 012 entries), twin/sexy prime catalogs, and statistical analysis code.