Cyclotomic Trace–Norm Rigidity: An Abstract Framework and a Finiteness Theorem

Let K = ℚ(ζ_N) be a cyclotomic field, let L ⊂ K be a quadratic subfield, let δ ∈ 𝒪_K \ {0}, and let U ⊂ K× be a finitely generated multiplicative subgroup. We study the relative traces β(u) = Tr_{K/L}(uδ), u ∈ U, after rational p-adic reduction and primitivization in L. The normalized traces define classes in L×/𝒪_L×. We prove that the complete normalized trace-class set is finite whenever [K : L] ≥ 2 and three independent hypotheses hold: non-degeneracy of the trace expansion, uniform finite support of the normalized traces, and a finite-intersection condition controlling the directions of U contained in L×. The proof converts the relative trace into a non-degenerate S-unit equation and then controls the fibers of the resulting ratio vectors. The norm-square condition is an additional Diophantine filter on an already finite class set. The theorem concerns scalar multiplicative traces with uniform normalized support. Affine or coupled selectors, variable-support trace systems, direct norm identities, and other Diophantine constructions require separate arguments and are outside the scope of this article.