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

We study the parametric family $p_{k,m,\epsilon ,q}=km(m+1)+\epsilon +2kq$ with $k,m\in {\mathbb{N}}^{\ast }$, $\epsilon \in \{\pm 1\}$, $q\in \mathbb{Z}$, which extends the elementary observation $p\equiv \pm 1(\mathrm{m}\mathrm{o}\mathrm{d}6)$ for every prime $p>3$. Each prime $p>k+1$ has a canonical triple $(m,\epsilon ,q)$ with $\mathrm{\mid }q\mathrm{\mid }$ minimal, mapping it to a generalised hexagonal base $H_m^{(k)}:=km(m+1)+1$ and a signed minimal offset $q_{\mathrm{m}\mathrm{i}\mathrm{n}}(k,m,\epsilon )$. Every quantitative assertion is labelled rigorous, conditional, or heuristic. \emph{(Rigorous, structural.)} For every prime $\mathrm{\ell }\mid 2k$ the family satisfies $p\equiv \epsilon (\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\ell })$: the degenerate modular axis is fixed exactly, independently of $q$. \emph{(Rigorous, constructive.)} For the $3$-smooth subfamily $p=3m(m+1)+1$ with $m=2^a{3}^b-1$, Pocklington–Lehmer certificates are unconditional; two a-priori arithmetic filters remove $\approx 87\mathrm{\%}$ of candidates before any large-integer arithmetic, and the algorithm produced an unconditionally certified prime of $29998$ 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 ($R^2=1.16\times {10}^{-7}$), 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.)} $\mathrm{\mid }{q}_{\mathrm{m}\mathrm{i}\mathrm{n}}\mathrm{\mid }=O_k(m{\log }^{2}m)$; $\mathbb{E}[\mathrm{\mid }q\mathrm{\mid }\mid m]=m\mathrm{/}4+O((\log m{)}^2\mathrm{/}\sqrt{m})$; a modular distribution law. \emph{(Conditional, RH.)} The per-prime $\zeta$-footprint on the layer occupancy is $\ll p^{-1\mathrm{/}4}(\log p{)}^2$ (Selberg variance) – the mechanism behind the negative result. \emph{(Heuristic.)} $\mathbb{E}[\mathrm{\mid }{q}_{\mathrm{m}\mathrm{i}\mathrm{n}}\mathrm{\mid }]\sim \log m\mathrm{/}{C}_k$ ($R^2=0.984$); the geometric constant $C_0(k)=⟨\mathrm{\mid }q\mathrm{\mid }⟩\mathrm{/}\sqrt{p}=1\mathrm{/}(4\sqrt{k})$, stable to $<0.02\mathrm{\%}$ across $3\le k\le 29$; and, for $\mathrm{\ell }\mid 2k$, $q_{\mathrm{m}\mathrm{i}\mathrm{n}}$ is empirically equidistributed modulo $\mathrm{\ell }$ (\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.