Even–Odd Energy Asymmetry in the Primorial Character Family and Connections to the Chebyshev Bias

We study the imaginary spectral energy E_Im = Σ c_χ² carried by the logarithmic derivatives −L'/L(1/2, χ) across the family of Dirichlet characters modulo the primorials d = 30, 210, 2310. The bridge operator decomposition M = H + iC separates each eigenvalue into an arithmetic component h_χ (depending only on conductor and parity) and a zero-dependent component c_χ = −2 Im[L'/L(1/2, χ)]. Decomposing E_Im by character parity reveals a dramatic asymmetry: odd characters carry 18.2 times more energy per character than even characters at d = 2310, with extreme concentration — 1.3% of characters account for 50% of the total energy. We trace this asymmetry to the Γ-factor dichotomy in the functional equation: odd characters carry Γ((s+1)/2), which permits zeros closer to s = 1/2 than the Γ(s/2) factor of even characters. The per-prime contributions to c_χ are shown to be quasi-independent, reflecting the Chinese Remainder Theorem factorisation of the character group. These findings provide a spectral explanation for the dominance of odd characters in the Rubinstein–Sarnak bias constants governing prime number races. We conjecture that c_max(d_k) = max_χ |c_χ| provides a quadratic lower bound on the logarithm of the first sign change in prime races at primorial moduli.