p-adic Distribution of the Minimal Offset q_min in Parametric Prime Families: Bateman–Horn Predictions, Conditional Theorems, and an Anti-Persistence Sign-Reversal Phenomenon
Year: 2026
Type: preprint
Abstract: We study the distribution modulo small primes ℓ of the minimal offset q_min(k, m, ε) for the parametric prime family p = km(m+1) + ε + 2kq, with k ≥ 1 and ε admissible (i.e., gcd(ε, 2k) = 1). This completes a trilogy of papers on this family: Paper I (April 2026) characterised the marginal law E[|q_min|] ~ log m / C_k; Paper II (April 2026, revised) established the conditional law E[|q| | m] = m/4 and excluded a spurious spectral connection to ζ(s); the present paper analyses the joint modular layer, completing the description with a third axis. Our principal contributions are: (i) An EXPLICIT FORBIDDEN RESIDUE q*(k, m, ε; ℓ) ∈ ℤ/ℓℤ (Proposition 2.1), giving rise to the PERMISSION FREQUENCY π_perm(r; k, ℓ). (ii) A CONDITIONAL THEOREM (Theorem 2.6): under H1' and H3', F_N(r; k, ℓ) = π_perm(r; k, ℓ) / (ℓ-1) + O(N^{-1/2}). Validated empirically at N = 10^6 for k ∈ {3, 7, 15, 21}: mean Pearson correlation prediction/observation r = 0.99 for small ℓ, with effective constant C̄ ≈ 0.5 (5× smaller than the triangle-inequality bound). (iii) An UNCONDITIONAL THEOREM (Theorem 2.8) for ℓ | 2k: F_N(r) = 1/ℓ + O(N^{-1/2}) unconditionally; verified across all 10 tested cases with χ² ≤ 3.1. (iv) A NEW EMPIRICAL PHENOMENON (Section 3): the lag-1 autocorrelation of (q_n mod 2) follows a clean LINEAR LAW in k, ACF_1 ≈ 0.024 k - 0.20 (R² = 0.97 across 10 values of k ∈ [3, 21] at N = 10^6), with sign-reversal threshold at k_0 ≈ 8.4. The full spectral signature {α(ℓ)} for ℓ = 1 to 10 of the slopes per lag is fitted by a DAMPED COSINE (period T ≈ 5.89, damping rate γ ≈ 0.39) to within R² = 0.999. The phenomenon AMPLIFIES with the modulus (factor ~2× between mod 2 and mod 5 at large k). At k = 21, mod 5: ACF_1 = +0.64. Connection to Paper II: Hypothesis H3' is a structural analogue of Paper II's H3 for the residue class. Theorem 2.6 thereby provides an INDIRECT QUANTITATIVE TEST of H3 via measurable distributions. The empirical agreement is consistent with H3 holding in the regime tested, but this constitutes empirical support, not proof: the unconditional barrier (parity problem; uniform PNT in arithmetic progressions of large moduli) is unchanged. Status of results: Theorem 2.8 is unconditional. Theorem 2.6, Proposition 6.1 (H3 ⇒ H3'), and the ACF conjectures are heuristic or conditional, with carefully tracked hypotheses. Section 6 discusses at length the obstacles to unconditional proof (Bateman–Horn open beyond the linear case; parity barrier blocking sieve approaches to degree ≥ 2 polynomials; uniform PNT in arithmetic progressions of large moduli unavailable). We are explicit that the empirical signature of Section 3 — the damped cosine fit at R² = 0.999 — may itself be the trace of a non-trivial second-order correlation that the Cramér–Granville hypothesis ignores.
Author: BAKKAOUI, HASSANE
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7160870965
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