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

Abstract

We study the parametric family $p_{k,m,\varepsilon,q}=km(m+1)+\varepsilon +2kq$ with $k,m\in\mathbb{N}^{\ast}$, $\varepsilon \in \{\pm1\}$, $q\in\mathbb{Z}$, which extends the elementary observation $p\equiv\pm1\pmod 6$ for every prime $p>3$. Each prime $p>k+1$ has a canonical triple $(m,\varepsilon,q)$ with $|q|$ minimal, mapping it to a generalised hexagonal base $H^{(k)}_m:=km(m+1)+1$ and a signed minimal offset $q_{\rm min}(k,m,\varepsilon)$. Every quantitative assertion is labelled rigorous, conditional, or heuristic. \emph{(Rigorous, structural.)} For every prime $\ell\mid 2k$ the family satisfies $p \equiv \varepsilon \pmod\ell$: the degenerate modular axis is fixed exactly, independently of $q$. \emph{(Rigorous, constructive.)} For the $3$-smooth subfamily $p=3m(m+1)+1$ with $m=2^a3^b-1$, Pocklington--Lehmer certificates are unconditional; two a-priori arithmetic filters remove $\approx 87\%$ of candidates before any large-integer arithmetic, and the algorithm produced an unconditionally certified prime of $29\,998$ decimal digits, independently re-verified in a separate computational environment. \emph{(Rigorous, negative.)} The earlier-reported spectral correlation between the cumulative functional $Q(r)=\sum q_n$ and the zeros of $\zeta$ is a spurious-regression artefact. Beyond three permutation tests and a bias-free test on $10^8$ primes ($R^2=1.16\times10^{-7}$), we prove \emph{unconditionally} that the regression amplitude $A_N(\gamma)\to 0$ for every fixed $\gamma$, by reduction to square-root-phase prime exponential sums. The Riemann zeros are not spectral frequencies of $Q(r)$. \emph{(Conditional, GRH / Bateman-Horn.)} $|q_{\rm min}|=O_k(m\log^2 m)$; $\mathbb{E}[|q|\mid m]=m/4+O((\log m)^2/\sqrt m)$; a modular distribution law. \emph{(Conditional, RH.)} The per-prime $\zeta$-footprint on the layer occupancy is $\ll p^{-1/4}(\log p)^2$ (Selberg variance) -- the mechanism behind the negative result. \emph{(Heuristic.)} $\mathbb{E}[|q_{\rm min}|]\sim\log m/C_k$ ($R^2=0.984$); the geometric constant $C_0(k)=\langle|q|\rangle/\sqrt p=1/(4\sqrt k)$, stable to ${<}0.02\%$ across $3\le k\le 29$; and, for $\ell\mid 2k$, $q_{\rm min}$ is empirically equidistributed modulo $\ell$ (\emph{not} a consequence of Bombieri--Vinogradov). This is experimental mathematics with rigorously tracked hypotheses: two genuinely unconditional pillars -- constructive (the certified prime) and analytic-negative (the vanishing $\zeta$-signal) -- together with a precise law on each axis. No classical question is settled.