Recurrence Implies Periodicity in Collatz-Type Maps — E8 Intelligence Research

FINDING: The arXiv paper (2601.03297v6) reveals a topological/ergodic framework where, for a class of maps including the Collatz function, recurrence within a specially constructed Borel σ-algebra implies periodicity — a structural bridge between measure-theoretic recurrence and number-theoretic cycles. | MATH: Let \(T: X \to X\) be the Collatz-type map on a suitable space \(X\) with a topology \(\tau\) generating Borel σ-algebra \(\mathcal{B}\). Key result: if \(x \in X\) is recurrent under \(T\) (i.e., for every neighborhood \(U\) of \(x\), \(\exists n>0\) with \(T^n(x) \in U\)), then \(x\) is periodic (i.e., \(\exists p>0: T^p(x)=x\)). No explicit constants or ratios appear; the core is the equivalence class of recurrence \(\Rightarrow\) periodicity under the chosen topology. | CONNECTION: The Borel σ-algebra is generated by open sets — a lattice-theoretic construction (complete Boolean algebra) whose atoms, if finite, mirror crystallographic point group orbits. The recurrence-perio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com