Collatz Conjecture via Cayley Graphs: Decidability and Cyclic Root Proof — E8 Intelligence Research

FINDING: Collatz conjecture remains unproven; recent work frames it as automorphic Cayley colour graph with decidability of $an+b$ conjectures, claiming a proof via root-path structure to cyclic root 4→2→1→4. | MATH: Collatz map $T(n) = n/2$ if n even, $3n+1$ if n odd; conjecture: ∀ n ∈ N, ∃ k: Tᵏ(n)=1. The arXiv paper (2008.13643v8) models Collatz as walks on a Cayley graph with branching number $c=4$, trivial cycle 4→2→1→4, and claims decidability of generalized $an+b$ conjectures. No explicit constants or ratios beyond $3,2,1,4$ appear in the abstract. | CONNECTION: The cyclic root 4→2→1→4 is a 3-cycle — a crystallographic rotation symmetry (order 3) in 2D lattice terms. The branching number $c=4$ relates to square lattice coordination number (4 nearest neighbours in Z²). The automorphic Cayley graph structure hints at group-theoretic symmetries, but no explicit golden ratio, base-60, or 0.618/0.786/1.618/2.618 consta Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com