Conductor decomposition of pair-count variance at fixed modulus: the c_pair(q) problem

The companion paper (DOI: 10.5281/zenodo.24055465) established that prime-class variance K_p is classical (Rubinstein–Sarnak, 1/log x decay, verified at three moduli) while pair-class variance K_K is scale-invariant and unexplained. This paper attacks the open problem: which characters carry the pair variance, and does it relate to the L-function zero spectrum? Three experiments and a two-modulus correlation test yield: (1) K_K's scale-invariance survives ΛΛ-weighting (the pair analogue of the ψ-test), closing the last bookkeeping escape hatch. (2) The primorial tower recurrence does not close over single primes — standalone K_K²(p) is flat at ~0.38 while fibre predictions climb to 0.51–0.74, proving c_pair must be built from conductor families, not prime factors. (3) A Parseval-exact conductor-family decomposition at three moduli (q = 2310, 30030, 510510) shows the highest-conductor family dominates (63% → 57% → 51%), pure-prime conductors vanish combinatorially (5.4% → 0.4% → 0.04%), and p = 3 is a spectator. (4) Per-character pair variance |D'(χ)|² shows no meaningful correlation with the single-character zero sums c(χ) = Σ_γ 1/(1/4 + γ²) in the admissible-class convention (r = 0.063, r² < 0.4%, z = 4.80 at n = 6), confirming that E₂(x, χ) = Σ Λ(n)Λ(n+h)χ(n) does not factor through individual L-function zeros. The brick wall predicted by the Goldston–Montgomery theory — no single governing L-function, no E₂ zero-sum — is now backed by data. ### Keywordsprime number theory; twin primes; Dirichlet L-functions; conductor decomposition; pair-correlation variance; Goldston-Montgomery; admissible classes; primorial modulus; scale invariance