Statistical Morphology and Geodesic Rigidity of Prime Constellations in Q(n) = n⁴⁷ − (n−1)⁴⁷: A Complete Census of the First 2 Billion Cases

We present an exhaustive morphological census of prime-generating integers n for the polynomial Q (n) = n⁴⁷ − (n−1) ⁴⁷ over the continuous domain n ∈ 1, 2×10⁹. The survey identifies 18, 473, 571 probable primes classified into four morphological types: 18, 121, 562 solitary primes, 173, 351 pairs, 1, 749 triplets, and 15 quadruplets (four consecutive integers each generating a probable prime of 341–430 digits). Key results: (1) Discovery of a 15th quadruplet at n = 23, 159, 557, previously absent from all published catalogs, recovered by filling the 10⁶, 10⁸ data gap. (2) Modular-3 Exclusion Principle: We prove that Q (n) ≡ 1 (mod 3) universally, rendering Q (n) +k composite for every offset k ≡ 2 (mod 3). This eliminates one-third of all candidate positions — including the right twin-prime slot k = +2 — establishing an intrinsic left–right chirality in the satellite field. (3) Geodesic Rigidity: The pair-to-solitary ratio R₂ (x) decays only ~7% over 0. 5×10⁹, 2×10⁹, significantly slower than the ~11% predicted by random independence models, confirming that constellation structure is arithmetically protected. (4) Asymptotic Validation: The observed ratio decay matches the Bateman–Horn prediction R (n) ~ K/ln (n) to within 1 percentage point (14% predicted vs. 15% observed over a factor-of-20 range). (5) Conditional Neighborhood Sieve: A formally defined search space S = Q (n) −k | n ∈ C₄, k ∈ 2, R, k ≢ 2 (mod 3) that exploits chirality and conditional density for deep-space surveys beyond n = 10¹¹. This repository contains the full paper (LaTeX source + compiled PDF), three publication-quality figures, the complete per-bin census data, the 15-quadruplet catalog, and the Python algorithms (exhaustive multi-core sweeper and chirality-aware deep-space radar) used in the survey.