The Collatz Conjecture: Open Despite Tao's Breakthrough and Graph-Theoretic Framing — E8 Intelligence Research

FINDING: Collatz conjecture remains open; strongest recent structural work models it as an automorphic Cayley colour graph, with Tao's partial results on almost-all numbers, but no proof of the full conjecture. | MATH: Collatz map T(n) = n/2 if n even, (3n+1)/2 if n odd (or 3n+1 then divide by 2). Conjecture: ∀n∈ℕ, ∃k: T^k(n)=1. Key known: Tao (2019) proved for almost all n (logarithmic density 1), but not all. The arXiv paper (2008.13643v8) frames the map as a walk on a Cayley graph with branching at c=4, trivial cycle 4→2→1→4. No new constants or closed-form ratios emerge from these sources. | CONNECTION: The Cayley graph framing hints at group-theoretic structure — but no explicit link to golden ratio (0.618, 1.618), base-60, or crystallographic root systems is given in the provided abstracts. The trivial cycle 4→2→1 has ratio 2 (doubling/halving), which is a dyadic symmetry, not the golden-ratio family. No 0.382, 0.786, or 2.618 appear. The automorphic colour graph suggests potenti Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com