We measure the count of k-term arithmetic progressions of primes as a function of the common difference Δ, on the 6N skeleton, sieving all primes to X = 10⁸ (5, 761, 455 primes). Viewed as a function of Δ, the count Nₖ (Δ) behaves exactly like a diffraction spectrum. (i) Selection rule (extinction): Nₖ (Δ) is identically zero unless Δ is divisible by the primorial Π≤₊ p (6 for k=3, 4; 30 for k=5). This is admissibility — if a prime p ≤ k does not divide Δ, the k terms cover every residue mod p, so one term is composite. We verify it as a literal zero: scanning every Δ ≤ 1512 for k=3, the number of off-lattice (6 ∤ Δ) progressions found is exactly zero. (ii) Resonance enhancement: on the admissible lattice, each additional prime q > k dividing Δ multiplies the count by the universal factor (q−1) / (q−k). This is the relative Hardy–Littlewood singular series 𝔖ₖ (Δ) = Πₚ (1 − νₚ (Δ) /p) / (1 − 1/p) ᵏ, with νₚ = 1 if p|Δ and min (k, p) otherwise; verified against the data to ≲1% across k = 3, 4, 5 (e. g. k=3 Δ=2310 → measured 3. 768 vs predicted 3. 750; k=4 Δ=210 → 8. 11 vs 8. 00; k=5 Δ=2310 → 4. 955 vs 5. 000). Both features are exact consequences of admissibility; what the experiment adds is the spectral presentation, the high-precision verification at X = 10⁸, and the unified per-prime resonance law (q−1) / (q−k) across k. We make no claim about the existence of arbitrarily long progressions: the Green–Tao theorem is the untouchable backdrop, not a target. As throughout this series, this is a measurement, not a theorem; the Hardy–Littlewood heuristic is taken as input. Part XXX of "Arithmetic Geodynamics on the 6N Skeleton. " Code and measured data: https: //github. com/Ruqing1963/6N-ap-resonance
Ruqing Chen (Mon,) studied this question.