A parametric family of primes p = km(m+1) + e + 2kq: heuristic laws, conditional theorems, and unconditional primality certificates

Abstract: We study the parametric family p_{k,m,e,q} = k m(m+1) + e + 2kq (k,m in N*, e in {+1,-1}, q in Z), extending the elementary fact that every prime p>3 satisfies p = +/-1 (mod 6). Each prime p>k+1 has a canonical triple (m,e,q) with |q| minimal; every assertion is tagged rigorous, conditional, or heuristic. Rigorous: (i) for every prime l dividing 2k, p = e (mod l) -- the degenerate modular axis is fixed exactly, independent of q; (ii) for the 3-smooth subfamily p=3m(m+1)+1 with m=2^a 3^b-1, Pocklington-Lehmer certificates are unconditional, two a-priori filters remove about 87% of candidates, and we exhibit an unconditionally certified prime of 29998 decimal digits, independently re-verified; (iii) the reported spectral correlation between Q(r)=sum q_n and the zeros of zeta is a spurious-regression artefact -- beyond three permutation tests and a bias-free test on 10^8 primes (R^2=1.16e-7), we prove unconditionally that the regression amplitude A_N(gamma) tends to 0 for every fixed gamma, by reduction to square-root-phase prime exponential sums. Conditional (GRH / Bateman-Horn / RH): |q_min| = O_k(m (log m)^2); E[|q| | m] = m/4 + O((log m)^2 / sqrt m); a modular distribution law; and a per-prime zeta-footprint << p^{-1/4} (Selberg variance). Heuristic (validated at 10^6-10^8): E[|q_min|] ~ (log m)/C_k, and the geometric constant C_0(k) = <|q|>/sqrt(p) = 1/(4 sqrt k), stable to <0.02% over 3<=k<=29. The work rests on two genuinely unconditional pillars: constructive (the certified prime) and analytic-negative (the vanishing zeta-signal). No classical question is settled; this is experimental mathematics with rigorously tracked hypotheses.