Prime Arithmetic Progressions as a Lattice Resonance: the Admissibility Selection Rule and the Singular-Series Enhancement Spectrum

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 = 10^8 (5,761,455 primes). Viewed as a function of Δ, the count N_k(Δ) behaves exactly like a diffraction spectrum. (i) Selection rule (extinction): N_k(Δ) is identically zero unless Δ is divisible by the primorial Πp≤k 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 𝔖_k(Δ) = Π_p (1 − ν_p(Δ)/p)/(1 − 1/p)^k, with ν_p = 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 = 10^8, 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