Proposition 12 ([Progressive characters]). Let J be finitely generated with the subgroup-field hypotheses, and let K J satisfy the selective realization property of Definition 8. Every finite family of rational expressions over J can, …

Take the preimage H_0 of a finite-index RFRS subgroup of J/K . Restriction of scalars is right multiplication on right-coset coordinates, using a normal core when necessary. The finite-index field decomposition makes this an exact finite matrix operation. Linearize the finitely many expressions as in Lemma 9 ([Controlled linearization]), retaining the specified expressions for all needed inverse entries. Pull back a normal RFRS chain from H_0/K . At stage i , use to expand each inverse matrix for both signs of a character χ_i with kernel N_i . Truncate sufficiently far to obtain the positive errors. Collect the finitely many coefficients of the matrix and its truncations; they lie in D(N_i) . Restrict to the next group by , and linearize these scalar problems. Lemma 10 ([Finite descent of inverse depth]) terminates this procedure with finitely many unresolved coefficients in D(K) . Choose a finite set of elements of K supporting rational expressions for all those coefficients. Apply the selective realization property, including its finite product of conjugate maps, to obtain a normal finite-index preimage H and a map injective on the required support subgroups. Pass to the preimage H_p of an RFRS subgroup of its image. This cut can mix coefficient entries, so we describe it explicitly. For a scalar c∈D(U_0) , group the cosets of H_p into orbits of right multiplication by U_0 . An orbit has representatives gu_j , with u_j∈U_0 . The entries of right multiplication by c in these coordinates come from decomposing u_jc over D(U_0∩g^ -1 H_p g) and then conjugating by g . They therefore admit finite rational expressions entirely in these subgroup fields. Changing to one common transversal only multiplies entries by monomials, which does not affect summability. Start the depth argument afresh with these new expressions. Use an RFRS chain in the new quotient and ordinary rational group polynomials. Work separately inside each support subgroup, choosing its own monomial transporters when taking a height coefficient. Later coefficients remain in conjugates of that support subgroup. Once a coefficient lies in the current character kernel, restriction to the next stage only conjugates it, because that whole kernel is contained in the next subgroup. Thus the maximal-depth ancestry argument applies exactly as before. The new quotient map is injective on each support subgroup, so its normal RFRS chain separates the finite nonidentity support lists. This time the terminal problems are ordinary group-ring polynomials, with no unresolved division-field coefficients. They are realized for the zero character if necessary. Now reconstruct the inverse problems backwards, through both finite chains and the selective switch, using Lemma 11 ([Recombination by perturbation]). At each step use both signs' residual lists and choose one sufficiently small positive perturbation. The result is a single character on a common finite-index subgroup and compatible realizations of all original expressions for both of its signs. For the chain-complex assertion, include the entries of a D(J) -chain contraction and all the inverse operands used in their expressions. Their exact identities persist under the common series embeddings, giving contractions over the two ordinary Novikov rings. Restriction of scalars does not change the underlying chain complex or its ordinary homology. If the final character were zero, its Novikov rings would be F , so these contractions would contradict the assumed nonzero ordinary degree-zero homology.