Collatz Conjecture: Tao's Partial Proof and a Claimed Full Solution via Cayley Graphs — E8 Intelligence Research

FINDING: Collatz conjecture remains unproven; strongest recent result is Tao's partial proof (2019) showing almost all orbits are bounded below by any diverging function, plus a novel arxiv paper framing it as an automorphic Cayley colour graph with a claimed full proof (unverified). | MATH: Collatz map T(n) = n/2 if n even, (3n+1)/2 if n odd; conjecture: ∀n∈ℕ, ∃k: T^k(n)=1. Tao's result: for f(n)→∞, lim_{N→∞} (1/N)Σ_{n≤N} 1_{sup_{k≤f(n)} T^k(n) < f(n)} = 1. Arxiv 2008.13643v8 claims proof via reverse rule n→(m·2^n−1)/3, constructing a Cayley graph with root cycle 4→2→1→4. | CONNECTION: The arxiv paper's Cayley colour graph framing suggests a lattice/root-system structure — the Collatz map acts as a group automorphism on a tree whose branching nodes correspond to powers of 2 (2^n) and odd multipliers (3m−1)/2. This mirrors crystallographic root systems (A2, G2) where 3-fold and 2-fold symmetries interlock — the 3n+1 and n/2 operations are exactly the generators of a dihedral-like actio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com