Parity Vectors and Dyadic Shifts in the Accelerated Collatz Map — E8 Intelligence Research

FINDING: The Collatz map's parity vector induces a binary expansion that conjugates the map to a dyadic shift on the 2-adic integers, with the accelerated map revealing paradoxical sequences and new bounds. | MATH: Accelerated Collatz: \(T(n) = (3n+1)/2\) for odd \(n\), \(T(n) = n/2\) for even \(n\). Parity vector \(v_k(n) = (n \bmod 2, T(n) \bmod 2, \dots, T^{k-1}(n) \bmod 2)\) maps to a binary expansion \(\sum v_i 2^i\). The map on 2-adics \(\mathbb{Z}_2\) is conjugate to the Bernoulli shift \(\sigma(x) = (x-1)/2\) (odd) or \(x/2\) (even) — a dyadic odometer. The arXiv paper (2605.13886) proves three theorems on paradoxical sequences (finite words that never appear as parity vectors) and bounds on stopping times, building on Terras (1976), Lagarias (1985), Tao (2019). | CONNECTION: The dyadic tree structure of the Collatz graph is a binary tree — each node branches by parity. This is isomorphic to the Stern–Brocot tree and to the Farey sequence, whose ratios converge to \(\phi = 1.61 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com