Collatz Conjecture via Cayley Graphs: Decidability and Cyclic Root Proof — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: 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 $4\to2\to1\to4$. | MATH: Collatz map $T(n) = n/2$ if $n$ even, $3n+1$ if $n$ odd; conjecture: $\forall n \in \mathbb{N}, \exists k: T^k(n)=1$. The arXiv paper (2008.13643v8) models Collatz as walks on a Cayley graph with branching number $c=4$, trivial cycle $4\to2\to1\to4$, 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 $4\to2\to1\to4$ 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 $\mathbb{Z}^2$). 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
Author: Caldin, Andrew Stewart
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204104457
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