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For existence, prescribe a compatible ray class action over K of moduli 2^m√-d , using principal ideals with generators having residues 1 at the 2's and a fixed nonsquare at the ramified prime. This fixes 2-power roots of unity by norm. K(i,√[4] 2 )/K is dihedral of order 8 (disjoint quadratic data); its maximal abelian subfield is 2-cyclotomic in the sense of being contained in K(μ_ 2^∞ ) . Thus its nontrivial commutator element can be prescribed compatibly with the ray condition. The √q 's are independently switchable by their new disjoint ramification. The full constraint field has finite commutator image over K , so the subgroup fixing it has Tate image containing a finite-index part of the closed commutator of the image over K . This includes an open subgroup in determinant one by the non-CM open-image theorem~. The coset therefore permits trace different from 2. After fixing such an element, Chebotarev to increasing precisions (degree-one places with chosen lifts) gives the sequence; in particular r_i→1 2-adically. The Sylow-2 quotient of Pic( O_ r_i ) is cyclic of order 2^ v_2(r_i-1) by the unit ratio at r_i and odd class number. Use its group-ring Galois character Ψ_ u,i with generator value 1+u . Its quadratic quotient is K(√r_i) (pull back the discriminant character by norm, trivial on the defining principal order relations). At the places over odd q N , Frobenius exponents in this variable are odd: for q^h=(x) , the ratio x/ x R has symbol (q/r)=-1 . At w, w the exponents have valuation exactly one. Indeed α R is square by Hilbert reciprocity on (α,π_i) , primary π_i at both 2's and units at other odd places. Thus α/α is a square with nonsquare root α/(√2)^h R ; both roots have the same symbol. Inertia at either place over r surjects onto the cyclic quotient, while Ψ_t(Fr_R)→1 since π_i is increasingly primary.
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