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 \emph{norm descent}: Dold congruences are functorial under finite extensions and preserved by prime–ideal norms $N_{K\mathrm{/}\mathbb{Q}}$, yielding integer ghosts from algebraic ones. We extend the theory from $\mathbb{Z}$ to Dedekind domains, and show that integrality is stable under both Hadamard and Witt products. Two rigidity theorems lie at the core: a \emph{cyclotomic residues theorem}, asserting that if the logarithmic derivative has only cyclotomic poles then integrality forces rationality; and a stronger \emph{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.