Collatz Conjecture via Automorphic Cayley Graphs and Decidability — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
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
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BibTeX
@misc{caldin2026collatz,
title = {{Collatz Conjecture via Automorphic Cayley Graphs and Decidability — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635465},
url = {https://zenodo.org/records/23635465},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Collatz Conjecture via Automorphic Cayley Graphs and Decidability — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635465
UR - https://zenodo.org/records/23635465
AB - 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 log_2 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
ER -