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.