p-adic Distribution of the Minimal Offset qₘin in Parametric Prime Families: Bateman–Horn Predictions, Conditional Theorems, and an Anti-Persistence Sign-Reversal Phenomenon

We study the distribution modulo small primes ℓ of the minimal offset qₘin (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|qₘin| ~ log m / Cₖ; 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 πₚerm (r; k, ℓ). (ii) A CONDITIONAL THEOREM (Theorem 2. 6): under H1' and H3', FN (r; k, ℓ) = πₚerm (r; k, ℓ) / (ℓ-1) + O (N^-1/2). Validated empirically at N = 10⁶ 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: FN (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 (qₙ mod 2) follows a clean LINEAR LAW in k, ACF₁ ≈ 0. 024 k - 0. 20 (R² = 0. 97 across 10 values of k ∈ 3, 21 at N = 10⁶), with sign-reversal threshold at k₀ ≈ 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: ACF₁ = +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.