We introduce an arithmetic framework for what we call cyclotomic trace–norm rigidity. Given a cyclotomic field K = ℚ (ζN), a quadratic subfield L ⊂ K, a fixed element δ ∈ 𝒪K, and a finitely generated multiplicative subgroup 𝒰 ⊂ K×—typically an S-unit group coming from a real subfield—we study invariants obtained by tracing the orbit uδ down to L and imposing a norm-square condition in L. After a natural p-adic reduction and primitivization, these invariants define classes in L×/𝒪L×. Our main theorem proves that, under a standard non-degeneracy hypothesis, assuming that the trace has length K: L ≥ 3, and imposing a finite-intersection condition excluding positive-rank directions of 𝒰 inside L× modulo global units, the resulting set of trace–norm classes is finite. The proof reduces the trace expression to a non-degenerate S-unit equation, invokes the finiteness theorem of Evertse–Schlickewei–Schmidt for unit equations, and then uses the finite-intersection condition to control the fibers from ratio vectors to classes in L×/𝒪L×. This Version 2. 1 makes explicit the construction of the fixed set of places S′ required for the reduction to an S′-unit equation. Starting from an initial support set S₀ for 𝒰, δ, and the places above p, we pass to its Gal (K/L) -saturation and adjoin every place of K above the uniform support set SL of the normalized traces. We prove that the trace summands, the trace itself, and all normalized ratios lie in the fixed group 𝒪×₊, ₒ′, independently of u. The p-primitive normalization is also stated using the rational p-content valuation νₚ (γ) = min🕌∣ ⌊v_𝔭 (γ) /e (𝔭/p) ⌋, which is valid in split, inert, and ramified cases. The revised proof treats the finitely many degenerate parameters separately and makes explicit the normalization argument over each fixed ratio vector. These additions complete the support and fiber arguments required by the finiteness theorem. We also formulate a refinement rigidity principle: passing from ℚ (ζN) to a cyclotomic refinement ℚ (ζM), with N ∣ M, does not create new trace–norm classes whenever no new essential quadratic layer, no new unit-rank input, and no new independent trace summands are introduced. Finally, we explain how the phenomena previously observed at levels N = 16—the refinement of level 8—and N = 18—the Eisenstein regime—fit as instances of the abstract framework.
Rogelio Méndez Ibarra (Tue,) studied this question.