The Exact Representation Kernel: Finiteness, Two-Way Equivalence and a Count-Level Bijection for the Symmetric Root-Bounded Relation Domain

For m ≥ 3 the finite root-bounded relation domain (Ω_m, Σ×Q_m, R_m) is an exact re-encoding of the reduced Goldbach count: Res(m) is non-empty iff R_G⁻(2m) > 0, #Res(m) = R_G⁻(2m), and Goldbach failure at 2m holds iff Ω_m is completely covered by the finitely many residue cells O_{m,σ,r}.