For an integer base b ≥ 2 let sb denote the base-b digit sum, and for a polynomial g in Zx consider the digit-sum map T₁, ₆ (n) = sb (g (n) ), iterated on the positive integers; its cycles include, as fixed points, the generalized Dudeney roots. We treat T₁, ₆ 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 Dₘax (b, g) and bound B (b, g), one has T₁, ₆ (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 T₁, ₊ (n) = sb (nᵏ) are the monomial case g (x) = xᵏ. Our main result is arithmetic: reduction modulo b−1 semiconjugates T₁, ₆ 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 T₁, ₆ 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 (nᵏ) 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 `cyclecensusb10. 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 cg = 0 used in Table 3. 6. Appendix A: `descentbound` 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.
Márcio Venício Pilar Alcântara (Sun,) studied this question.