Proposition 12 ([Finite-precision ring-class characters]). Fix finitely many split primes of K above odd rational primes, outside 2N , and orient one prime in each conjugate pair. There are distinct split rational primes r_i avoiding all …

We give the compatibility argument before imposing the Tate condition. Choose prime ideals p_1,…, p_s with pairwise disjoint conjugate pairs, including the oriented prescribed primes and the primes above 2N , and enough additional split prime ideals to generate Cl(K) . Such additional ideals may avoid any fixed finite set. Put weight w_j=1 at a prescribed odd prime and w_j=0 at every additional prime and every prime above 2N . The conjugate weight is -w_j . Let R=( Z^s(K)), choose a basis m^ (a) of R , and choose β_a∈K^× with (β_a)=∏_j p_j^ m_j^ (a) . Work first over F_∞=K(μ_ 2^∞ ) and adjoin 2^n -th roots of the finitely many β_a/β_a . Prescribe on these roots the multipliers ζ_ 2^n ^ \,-∑_j m_j^ (a) w_j . The minus sign fixes the choice of connecting map in the ring-class sequence and could be reversed throughout. These prescriptions respect every Kummer relation. Indeed, if ∏_a(β_a/β_a)^ b_a ∈(F_∞^×)^ 2^n , take its valuation at a prime above each p_j of nonzero weight. These odd primes are unramified in F_∞/K . Thus 2^n divides ∑_a b_am_j^ (a) for each such j , proving the required compatibility of. Primes over 2 have weight zero and impose no division by a ramification index. Units impose no additional weighted relation. Kummer duality supplies the prescribed automorphism of the radical extension over F_∞ . At a rational prime r split in a sufficiently large finite cyclotomic subfield, the power-residue character of F_r^× is surjective onto μ_ 2^n . The split ring-class exact sequence is 1⟶ ( O_K/r O_K)^×/( Z/r Z)^× ( Z+r O_K) (K)⟶1. There is no unit correction under our exclusion of the extra-unit fields. The first group is F_r^× , with a principal relation measured by β/β at the oriented prime over r . Consequently says exactly that its primitive power-residue character extends to the class group with the specified values w_j . Since the ring-class extension is unramified at 2N , value zero on Frobenius means that the resulting local character is trivial there. Conjugation acts by inversion. We now show that imposing these finitely many radical conditions cannot force either forbidden Tate trace. In the non-CM case the image of G_ K(μ_ 2^∞ ) is an open subgroup of SL_2( Z_2) , by the open-image theorem . Its index in the corresponding fixed- E image changes by at most two when K changes. Its commutator subgroup contains a fixed open subgroup U_E of SL_2( Z_2) . One may see uniformity by starting with a principal congruence subgroup in the fixed image: every index-two subgroup contains its squares, and the commutators of the resulting fixed congruence subgroup are open. Because the radical extension over F_∞ is abelian, these commutators act trivially on it. Hence every permitted Tate coset contains a translate of U_E . For completeness, in the CM case let M be the CM field. The image over KM(μ_ 2^∞ ) contains a fixed open subgroup of the norm-one Cartan; the index changes by a bounded factor, as [KM: Q]≤4 . Choose γ∈G_ KM with χ_ cyc (γ)=5^4=625 . This is possible uniformly: the cyclotomic image of G_ KM has index at most four in Z_2^× , so contains fourth powers. On the radical Galois group, conjugation by γ is raising to 625 , because all the radicands lie in K . On the Tate image over KM , conjugation is trivial. Therefore γuγ^ -1 u^ -625 , u∈G_ KM(μ_ 2^∞ ) , fixes every prescribed radical and acts on T as ρ_T(u)^ -624 . Raising to the fixed nonzero power 624 carries an open subgroup of the one-dimensional norm-one Cartan onto an open subgroup. This proves the required uniform kernel statement also in the CM case. The open Cartan image used here is the CM part of the usual ℓ -adic image theorem . On a coset of an open subgroup of SL_2 , trace is not identically 2 or -2 . The same holds on a coset of an open norm-one Cartan subgroup; on the other Cartan-normalizer component trace is zero. The relevant determinant-one sets are compact and have finitely many cosets modulo the fixed open subgroup. Choose in each coset an element whose trace is neither forbidden value. The largest of the finitely many resulting valuations is a constant depending only on E . Finally restrict the prescribed radical automorphism and such a Tate lift to a finite Galois extension, including as many roots of unity as required. Chebotarev gives infinitely many rational primes with this Frobenius class; choose the place over the prime to realize the chosen representative. They split in K and have arbitrarily high prescribed congruence to 1 . Since (1_r)=1±a_r+r , a congruence precision larger than the fixed trace bound gives all of. This is a finite Chebotarev condition at each stage, not a demand that one prime split in an infinite extension.