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

Technical Summary: A Parametric Family of Primes

Problem Statement
The paper investigates the distribution of prime numbers through a specific parametric family that generalizes the elementary observation that every prime $p > 3$ satisfies $p \equiv \pm 1 \pmod 6$. The authors define a family of integers:
$$p_{k,m,\varepsilon,q} = km(m+1) + \varepsilon + 2kq$$
where $k, m \in \mathbb{N}^*$, $\varepsilon \in \{\pm 1\}$, and $q \in \mathbb{Z}$. For a fixed prime $p$, the authors identify a unique "canonical triple" $(m, \varepsilon, q)$ such that $|q|$ is minimized. The central object of study is this minimal offset, $q_{\min}(k, m, \varepsilon)$, and its behavior across three layers: marginal (size vs. $m$), conditional (distribution given $m$), and modular (distribution modulo $\ell$). The study aims to distinguish between rigorous structural laws, conditional theorems (assuming GRH, RH, or Bateman–Horn), heuristic predictions, and negative results regarding spectral correlations.

Methodology
The research employs a multi-layered approach combining analytic number theory, computational number theory, and statistical analysis:

1. Structural Analysis: Rigorous proofs using elementary number theory and Dirichlet's theorem on arithmetic progressions to establish exact modular constraints and bounds.
2. Conditional Theorems: Derivation of bounds and distribution laws assuming standard conjectures (Generalized Riemann Hypothesis, Riemann Hypothesis, Bateman–Horn conjecture).
3. Computational Construction: Implementation of the "PrimeQuest" algorithm to generate and certify large primes within the 3-smooth subfamily ($k=3, \varepsilon=1, m=2^a 3^b - 1$) using the Pocklington–Lehmer criterion.
4. Statistical Refutation: Application of permutation tests (shuffle, block, Theiler surrogates) and bias-free regression analysis on $10^8$ primes to test for spectral correlations between the cumulative functional $Q(r) = \sum q_n$ and the zeros of the Riemann zeta function.
5. Analytic-Negative Proof: Reduction of the spectral correlation question to square-root-phase prime exponential sums, proving unconditionally that any linear signal from zeta zeros vanishes.

Key Contributions and Results

• Rigorous Structural Laws:

  • Degenerate Modular Axis: The paper proves that for any prime $\ell$ dividing $2k$, every prime in the family satisfies $p \equiv \varepsilon \pmod \ell$ exactly, independent of $q$. This fixes a "degenerate" modular axis.
  • Unconditional Certificates: For the subfamily $p = 3m(m+1)+1$ with $m = 2^a 3^b - 1$, the authors prove that Pocklington–Lehmer certificates are unconditional. They constructed a prime of 29,998 decimal digits ($m^* = 2^{19435} \cdot 3^{19173} - 1$) and certified it in under 4 hours on consumer hardware.
  • Arithmetic Filters: Two a-priori filters (quadratic reciprocity and mod-7 congruence) were proven to eliminate approximately 87.1% of candidates before any large-integer arithmetic is required.

• Conditional Theorems:

  • Bounds on $q_{\min}$: Assuming the Generalized Riemann Hypothesis (GRH), the authors prove $|q_{\min}| = O_k(m (\log m)^2)$.
  • Conditional Mean: Under GRH and Bateman–Horn hypotheses, the conditional expectation is derived as $E[|q| \mid m] = m/4 + O((\log m)^2/\sqrt{m})$.
  • Zeta Footprint: Assuming the Riemann Hypothesis (RH), the per-prime "footprint" of zeta zeros on layer occupancy is bounded by $\ll p^{-1/4}(\log p)^2$.

• Heuristic Laws (Validated Numerically):

  • Marginal Law: Empirical data ($N=10^6$) supports $E[|q_{\min}|] \sim \log m / C_k$, where $C_k$ is a constant.
  • Geometric Constant: A universal geometric identity $C_0(k) = \langle |q| \rangle / \sqrt{p} = 1/(4\sqrt{k})$ is observed to be stable to within 0.02% across tested $k$.
  • Gap Law: For $k=3$, the mean gap follows a Hardy–Littlewood singular series law.

• Negative Results (Refutation of Spectral Correlation):

  • The paper rigorously invalidates earlier claims of a spectral correlation between the cumulative functional $Q(r)$ and the zeros of $\zeta(s)$.
  • Statistical Evidence: At $N=10^8$, regression amplitude $R^2$ drops to $1.16 \times 10^{-7}$, and permutation tests confirm the absence of significant signal.
  • Analytic Proof: Theorem 6.6 proves unconditionally that the regression amplitude $A_N(\gamma) \to 0$ as $N \to \infty$ for any fixed $\gamma$. The zeros of $\zeta$ are shown not to be spectral frequencies of $Q(r)$; their influence is limited to statistical constraints at the scale of $\sqrt{x}$ (Selberg variance), not linear spectral signals.

Significance and Claims
The paper explicitly states that it does not settle any classical open problems in number theory (such as the parity barrier, the Riemann Hypothesis, or the uniform Prime Number Theorem at $N^{1/2}$). Instead, its significance lies in:

1. Epistemic Rigor: It introduces a framework for "experimental mathematics" where every assertion is strictly tagged as Rigorous, Conditional, Heuristic, or Negative.
2. Constructive Achievement: It provides an unconditional, independently verified record of a 29,998-digit prime generated via a specific parametric form.
3. Analytic Clarification: It decisively closes the question of a direct spectral link between prime offsets and zeta zeros, proving that such correlations are spurious regression artifacts.
4. Mapping a Territory: It defines a precise, falsifiable structure for studying prime distribution through parametric families, offering a "complementary direction of study" rather than a solution to existing conjectures.

The authors emphasize that the geometric constant $C_0(k) = 1/(4\sqrt{k})$ is a derived identity, not a mysterious invariant, and that the "negative" result regarding zeta zeros is a decisive, unconditional theorem derived from the vanishing of specific exponential sums.