Digit-Sum Power Maps: A Critical Survey of Cycles, Fixed Points and Arithmetic Dynamics

A critical survey of the iterative dynamics of the digit-sum power maps Tb,k(n) = s_b(n^k), where s_b is the sum of the base-b digits. The object is fixed precisely and separated from the digit maps with which it is routinely confused: the digitwise "happy"/perfect-digital-invariant maps, the digital root, additive persistence, and the analytic study of s_b(n^k) as an equidistributed sequence. The paper states a reproducible search protocol and records its outcome as a bibliographic matrix classified by proximity to the object; it organizes the known results into a six-family taxonomy (bounds, descent, eventual periodicity, cycles, basins, computation), attributing each result to a primary source and distinguishing theorem from conjecture, from finite enumeration, from informal observation; and it delimits the open problems. The central finding is that, although the fixed points (the generalized Dudeney roots) are well documented and the adjacent digitwise and analytic literatures are mature, the full cycle structure and the basins of attraction, with completeness proofs valid for all (b,k), have received almost no systematic peer-reviewed treatment. This gap is the agenda of the companion research paper. Version 2 (July 2026). Attribution and cross-reference updates; no critical finding changed and the gap table stands: 1. OEIS trajectory records added (Sections 6.2 and 8, bibliographic matrix, new reference). Sequences A056527 and A056529 (H. Bottomley, June 2000) and A061905(A. Auel, 2001) record empirically the trajectories, destinations and 2-cycle of the decimal square map — precisely the residue-class content of the arithmetic-basins theorem for k = 2. They now appear in the cycles section, in the basins section (with the remark that the theorem supplies the proof of what the OEIS recorded), and as a Class-A row of the bibliographic matrix.2. Reference [1] updated to Version 3 (all-versions DOI 10.5281/zenodo.19240836), noting that the current release incorporates an absorbing-set lemma, states the sharper cycle bound as an explicit corollary, and cites the OEIS antecedents.3. The companion paper is now published: reference [2] updated with the final title and DOI (all versions) 10.5281/zenodo.21348126, and verb tenses adjusted throughout (the results it announces are published, no longer forthcoming).4. Currency note (Section 6.1): the fixed-point layer remains actively curated — in June 2026 M. F. Hasler added to OEIS A046000 an empirical upper bound and the digit-count argument behind it.5. Minor wording: "independent observation" in the Acknowledgments; a capitalization typo fixed in the Conclusion.