The Dedekind Bridge — From Steinbach to Spectral Gap via ζ_Q(ρ)

We construct the Dedekind zeta function of the Steinbach field Q(ρ), where ρ = 2cos(π/7) is the shorter diagonal of the unit heptagon, and evaluate it at negative integers. Six results emerge, all derived from established number theory. First, the structural ladder: ζ(−3) = 1/120 = 1/|2I| (universal), while ζ_Q(√5)(−3) = 1/60 = 1/|A₅|, with L(−3, χ₅) = 2 matching the kernel order of the double cover 2I → A₅. Second, the spectral gap λ₁ = 168 on S³/2I encodes the heptagon through the McKay correspondence: 7 is the first non-trivial exponent of E₈. Third, disc(Q(ρ)) = 49 = 7² forces class number h = 1 via the Minkowski bound. Fourth, E₈ exponents sum to 120 = |2I|, with 7 at the foundation. Fifth, the full…