Collatz Conjecture via Automorphic Cayley Graphs and Decidability — E8 Intelligence Research

FINDING: A claimed proof of the Collatz conjecture via automorphic Cayley colour graphs, where the $3n+1$ map is encoded as a group action on a rooted tree, making the conjecture equivalent to a decidability statement about $an+b$ generalizations. | MATH: Collatz map $T(n) = n/2$ if n even, $T(n) = 3n+1$ if n odd; the "trivial cycle" 4 → 2 → 1 → 4; the paper (arXiv:2008.13643v8) constructs a Cayley graph whose automorphism group encodes T; the conjecture becomes: every vertex has a path to the cycle containing $c=4$. The $an+b$ conjecture generalizes $3n+1$ to $an+b$ with $a,b$ coprime, $a>1$. | CONNECTION: The Cayley graph structure is a lattice/root-system-like object — the branching at odd numbers (multiplying by 3) and halving at even numbers creates a binary tree with a single cycle. The ratio of odd-to-even steps in the Collatz trajectory statistically approaches log₂ 3 ≈ 1.585, which is *not* a golden-ratio harmonic, but the *automorphism group* of the Ca Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com