Lemma 71 ([Realization of labels and edges]). There is a linear map ξ:H^*→V such that loc_S(p_i^*)=ξ(l_i) , and the map (z,y)_S(z)+ξ(y), H⊕H^*⟶V, is injective. Arbitrary finite label lists and edge values are realizable by distinct fresh …

Quadratic reciprocity determines the localization of p_i^* at each odd place of S . At 2 , its squareclass is determined by the symbol of 2 , since p_i^*≡14 ; its real sign is determined by the symbol of -1 . These descriptions are linear in l_i , giving ξ . Global Hilbert reciprocity gives _S(z),ξ(y)⟩=y(z) (z∈H, y∈H^*). The subspace loc_S H is isotropic: outside S , both arguments are odd local units and their Hilbert symbols are trivial. If loc_S(z)+ξ(y)=0 , pairing with this subspace first gives y=0 , and valuations and sign then give z=0 . Notice also that V=2m . The displayed edge relation is quadratic reciprocity in signed-prime notation. Choose the primes successively. Their labels prescribe congruence classes at the fixed support, while the edges to earlier primes prescribe additional quadratic-residue classes. The Chinese remainder theorem and Dirichlet's theorem realize these prescriptions; reciprocity then determines precisely the reverse edges. This also realizes every label list.