A Parametric Family of Primes p = km(m+1) + ε + 2kq: Heuristic Laws, Conditional Theorems, and Unconditional Primality Certificates

This preprint undertakes a systematic study of the parametric family ofintegers p = km (m+1) + ε + 2kq, which extends the elementary observationp ≡ ±1 (mod 6) for every prime p > 3. Each prime p > k+1 admits acanonical representation as a triple (m, ε, q) with |q| minimal, mappingit to a generalised hexagonal base Hₘ^ (k): = km (m+1) +1 and a signedminimal offset qₘin (k, m, ε). The principal results, carefully labelled as rigorous, conditional, orheuristic, are: — Theorem A (Rigorous): for every prime ℓ dividing 2k, qₘin is uniformly distributed modulo ℓ with error O (N^-1/2), unconditionally on Bateman–Horn hypotheses, by reduction to Bombieri–Vinogradov at fixed modulus. — Theorem B (Rigorous): for the 3-smooth subfamily p = 3m (m+1) +1 with m = 2ᵃ · 3ᵇ − 1, Pocklington–Lehmer certificates are produced unconditionally. The algorithm has produced an unconditionally certified prime of 29, 998 decimal digits, independently re-verified in a separate computational environment. — Theorems C, D, E (Conditional): under GRH and Bateman–Horn type hypotheses, quantitative laws on the size, conditional expectation, and modular distribution of qₘin. — Heuristic Laws H1, H2, H3 (validated numerically at N = 10⁶ to 10⁸): marginal logarithmic law E|qₘin| ~ log m / Cₖ; universal geometric constant C₀ (k) = 1/ (4√k) ; Hardy–Littlewood gap law. — Decisive Negative Result: earlier-reported spectral correlations between cumulative functionals Q (r) and the zeros of ζ (s) are statistical artefacts of integrated-process regression. Three independent permutation tests (shuffle, block, Theiler surrogate) and a bias-free increment test on 10⁸ primes (R² = 1. 16 × 10^-7, z = -2. 20σ) establish their absence. This work is positioned as experimental mathematics with rigorously trackedhypotheses.