The Hydra Map and Numen Formalisms for Collatz-Type Problems

This paper details a generalization of the formalism presented in the author’s 2024 paper, “The Collatz Conjecture and Non-Archimedean Spectral Theory - Part I - Arithmetic Dynamical Systems and Non-Archimedean Value Distribution Theory”, to the case of Hydra maps on the ring of integers $\mathcal{O}_{K}$ of a global field $K$. In addition to recounting these definitions, background material is presented for the necessary standard material in algebraic number theory and integration and Fourier analysis with respect to the $p$-adic Haar measure. This paper is meant to serve as a technical manual for use of Hydra maps and numens in future research.

💡 Research Summary

The manuscript “The Hydra Map and Numen Formalisms for Collatz‑Type Problems” develops a comprehensive framework for studying a broad class of generalized Collatz maps—called Hydra maps—on the ring of integers (\mathcal O_K) of an arbitrary global field (K). Building on the author’s 2024 work, the paper first establishes a uniform notation that simultaneously handles archimedean (real/complex) and non‑archimedean (p‑adic) completions of (K). Standard concepts such as places, absolute values, valuations, and completions are reviewed, and a non‑standard congruence notation (x\equiv y) and a convergence notation (K=) are introduced to streamline later arguments.

A Hydra map (H) is defined by a piecewise rule that combines a linear (F)-linear transformation (r\in\operatorname{End}F(K)) with a constant shift (c\in\mathcal O_K) and a scaling factor determined by the valuation (v{\mathcal P}(x)) with respect to a finite set (\mathcal P) of prime ideals of (\mathcal O_K). Explicitly, \