Digit-Sum Maps of Polynomial Generators over Arbitrary Bases: Global Descent, Cycle Localization and Arithmetic Basins

For an integer base b ≥ 2 let s_b denote the base-b digit sum, and for a polynomial g in Z[x] consider the digit-sum map Tb,g(n) = s_b(g(n)), iterated on the positive integers; its cycles include, as fixed points, the generalized Dudeney roots. We treat Tb,g as a discrete dynamical system, uniformly in the base b and the generator g. For the admissible generators (integer polynomials of degree ≥ 2 with positive leading coefficient) we prove an effective global descent theorem: with an explicit threshold D_max(b,g) and bound B(b,g), one has Tb,g(n) < n for n > B(b,g), and {1,…,B(b,g)} is forward invariant and absorbing; hence every orbit is eventually periodic and every cycle lies in this finite interval, reducing the classification to a finite exact-integer computation. The power maps Tb,k(n) = s_b(n^k) are the monomial case g(x) = x^k. Our main result is arithmetic: reduction modulo b−1 semiconjugates Tb,g to the reduced generator ḡ(x) = g(x) mod (b−1) on Z/(b−1)Z, inducing a map γ on cycles that is always surjective; and if γ is injective, then every basin of attraction is a union of residue classes modulo b−1, with natural density |image basin|/(b−1) — γ being injective iff Tb,g has as many cycles as ḡ on Z/(b−1)Z. The decimal square map s(n²) has three attractors {1}, {9}, {13, 16}, with basin densities 2/9, 1/3, 4/9; this residue-class structure, recorded empirically in the OEIS since 2000 (A056527, A056529; also A061905), is here proved and explained. The phenomenon is base-dependent and fails, e.g., for s(n^k) with k ≥ 3. We give worked polynomial examples, entry-time bounds, a complete decimal cycle census for 2 ≤ k ≤ 60 — proved complete by the localization theorem, with machine-readable data and exact-integer code in the supplementary reproducibility archive. This is the second part of a two-part program: the first part is the critical survey "Digit-Sum Power Maps: A Critical Survey of Cycles, Fixed Points and Arithmetic Dynamics" (DOI, all versions: 10.5281/zenodo.21348064). Version 2 (July 2026). Corrections and attribution improvements; no result changed (all tables were independently re-verified in exact integer arithmetic): 1. Section 9: the maximal cycle length for 2 ≤ k ≤ 60 is 12, attained at k = 41, as Table 4 and the reproducibility data already recorded; the prose erroneously said 11.2. Census scope clarified: Table 4 tabulates 2 ≤ k ≤ 41; the rows up to k = 60 are in `cycle_census_b10.csv` (reproducibility archive) and in Figure 1.3. OEIS antecedents of the decimal square map added: A056527 and A056529 (H. Bottomley, 2000) and A061905 (A. Auel, 2001) record empirically the trajectory classification that Corollary 7.1 proves and Theorem 6.1 explains (new Remark in Section 7.1; reference [3] updated; the novelty claim in Section 1.2 qualified accordingly).4. Reference [1] now cites Version 3 of the companion preprint via the all-versions DOI.5. A one-line justification was added for the sharper constant c_g = 0 used in Table 3.6. Appendix A: `descent_bound` now certifies that the search cap exceeds the Lemma 3.3 bound.7. The companion survey ("Digit-Sum Power Maps: A Critical Survey of Cycles, Fixed Points and Arithmetic Dynamics") is now cited: the literature-search attribution in Section 1.2 was corrected to point to it, and the paper is identified as the second part of the two-part program.