Dold-Gauss Congruences, Norm Descent, and Rational Rigidity

We develop a Witt--Hadamard calculus for Euler products that unifies the classical Gauss congruences with their modern refinement, the Dold congruences. Within this framework we prove norm descent: Dold congruences are functorial under finite extensions and preserved by prime--ideal norms NK/Q, yielding integer ghosts from algebraic ones. We extend the theory from Z to Dedekind domains, and show that integrality is stable under both Hadamard and Witt products. Two rigidity theorems lie at the core: a cyclotomic residues theorem, asserting that if the logarithmic derivative has only cyclotomic poles then integrality forces rationality; and a stronger Dold⁺ rigidity theorem, showing that any algebraic series satisfying refined Dold congruences is necessarily rational. These results sharpen the Gauss--Dold picture: ordinary congruences enforce integrality, while the strengthened form collapses algebraic cases to rational ones. Applications include prime--ideal ladders in number fields and exact product laws for dynamical zeta functions, illustrated for subshifts of finite type and circle doubling.