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

We introduce an arithmetic framework for what we call cyclotomic trace--norm rigidity. Given a cyclotomic field K = Q(zeta_N), a quadratic subfield L ⊂ K, a fixed element δ ∈ O_K, 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^×/O_L^×.

Our main theorem proves that, under a standard non-degeneracy hypothesis and assuming the trace has length [K:L] ≥ 3, the resulting set of trace--norm classes is finite. The proof reduces the trace expression to a non-degenerate S--unit equation and invokes the finiteness theorem of Evertse--Schlickewei--Schmidt for unit equations.

We also formulate a refinement rigidity principle: passing from Q(zeta_N) to a cyclotomic refinement Q(zeta_M) 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 (refinement of level 8) and N = 18 (Eisenstein regime) fit as instances of the abstract framework.