Attracting fixed points of Dedekind zeta functions

Every Dedekind zeta function has a unique fixed point on $(1,\infty)$, but its stability and real two-cycles depend on arithmetic data. We give certified attracting examples in quadratic, cyclic cubic, cyclic quartic and cyclotomic fields. In contrast, infinitely many real and imaginary quadratic fields have a repelling fixed point and at least two distinct real two-cycles, disproving arithmetic rigidity. Explicit sufficient conditions require splitting at $2$ and inertness at every odd prime up to $2^{100}$. Three quadratic Euler-product limits explain both attraction and this coexistence; their missing pole creates a boundary layer absent from compact convergence on $s>1$. Prime-power cyclotomic zeta functions tend to $(1-p^{-s})^{-1}$, with explicit thresholds for attraction. We also prove that quadratic attraction has a positive natural density, the same for either sign of the fundamental discriminant and equal to a random-Euler-product probability. The moving multiplier has a non-atomic distribution. The exact absence-of-two-cycles criterion is a global sign condition on the second iterate, not merely repulsion of the fixed point. Numerical enumerations and density estimates are distinguished from proofs and interval-arithmetic certificates. MSC 2020:Primary 11R42, 37E05; Secondary 11M06, 11R11, 11R18. Version 1.1.0 (2026-09-05): the arithmetic-rigidity conjecture of v1.0.0 is replaced by Theorem F, its refutation — quadratic fields with a repelling fixed point and two pairs of real two-cycles, infinitely many real and imaginary, via the pole-free limit zeta(2s)(1+2^-s)/(1-2^-s). Supplement to Theorem A: certified cyclic examples (a cyclic cubic field of conductor 235002879 with an attracting fixed point; cyclic quartic fields of conductors 208 and 4528) with scan statistics for cyclic fields of degree 3, 4, 5, 7. Archive contents (zip): LaTeX source, abstract, certificates and exact checkers with logs/manifests/provenance, scan snapshots, proof notes, change log. See README.md and CHECKSUMS.txt.