Second moments of primes and prime pairs at fixed modulus: Rubinstein–Sarnak scaling for primes, scale-invariance for pairs

We study the variance of prime and prime-pair counts across residue classes modulo a fixed integer q, focusing on primorial moduli q = P_n (the product of the first n primes) for n = 5, 6, 7, at three computational scales (10^8, 10^9, 10^10). The index of dispersion is K = sqrt(Var/E), where Poisson corresponds to K = 1. Through a three-way comparison — real primes, a Cramér random model, and a Granville sieve-weighted model — we find: (1) The Markov-predicted variance K_stoch ≈ 0.958 is gap-invariant and reproduced by the Granville model. (2) The prime-class variance K_p ≈ 0.37–0.68 is sub-Poisson, scales as 1/sqrt(log x), and is quantitatively explained by Rubinstein–Sarnak zero-sums. We verify this at three moduli by computing C(q) = Σ_χ Σ_γ 1/|ρ_χ|² over Dirichlet L-function zeros and comparing against empirical variance constants. The three-modulus offset is ~5%, identified as a measured (0.73)/log x partial-summation conversion from Λ-weights to π-weights; the Λ-weighted (ψ-weighted) statistic is offset-free, matching zero-sums to <1%. (3) The pair-class variance K_K ≈ 0.59–0.87 is sub-Poisson and scale-invariant: it shows no measurable 1/log x decay across three orders of magnitude in x. It is universal across 47 gap types spanning six admissibility classes, and grows with the primorial tower in a manner that depends on the specific prime-divisor set of q — not on any single aggregate variable (ω(q), log log q, A/φ, φ/q). A tower decomposition shows 92–94% of K_K² at each level comes from newly introduced characters; a non-primorial test (q = 630 vs. q = 210) proves the sub-Poisson suppression is governed by L-function character structure, not local sieve admissibility. The prime variance is classical; the pair variance is the open problem.