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 H_m^(k) := km(m+1)+1 and a signedminimal offset q_min(k, m, ε). The principal results, carefully labelled as rigorous, conditional, orheuristic, are: — Theorem A (Rigorous): for every prime ℓ dividing 2k, q_min 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 = 2^a · 3^b − 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 q_min. — Heuristic Laws H1, H2, H3 (validated numerically at N = 10^6 to 10^8): marginal logarithmic law E[|q_min|] ~ log m / C_k; universal geometric constant C_0(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 10^8 primes (R² = 1.16 × 10⁻⁷, z = -2.20σ) establish their absence. This work is positioned as experimental mathematics with rigorously trackedhypotheses.