Cayley Graph Proof of the Collatz Conjecture via Decidable an+b Problems — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The Collatz map is reformulated as an automorphic Cayley colour graph, yielding a proof of the 3n+1 conjecture via decidability of an+b conjectures. | MATH: Collatz map T(n) = n/2 if n even, T(n) = 3n+1 if n odd; trivial cycle 4 → 2 → 1 → 4; branching number c=4; automorphism group of the Cayley graph acts transitively on root paths; decidability of an+b conjectures reduces to graph automorphism properties. | CONNECTION: The branching number c=4 and the cyclic root 4→2→1→4 form a 4-cycle — a square, the simplest crystallographic lattice in 2D (square lattice symmetry p4m). The ratio of the cycle's edge lengths (if mapped to a geometric square) is 1:1, but the *walk lengths* to the root from any n scale logarithmically — the inverse of the golden ratio appears in the density of odd numbers in the Collatz sequence: for a random odd n, the expected number of divisions by 2 before next odd is $2$, giving a contraction factor of 3/4 per o
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026cayley,
title = {{Cayley Graph Proof of the Collatz Conjecture via Decidable an+b Problems — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635651},
url = {https://zenodo.org/records/23635651},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cayley Graph Proof of the Collatz Conjecture via Decidable an+b Problems — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635651
UR - https://zenodo.org/records/23635651
AB - FINDING: The Collatz map is reformulated as an automorphic Cayley colour graph, yielding a proof of the 3n+1 conjecture via decidability of an+b conjectures. | MATH: Collatz map T(n) = n/2 if n even, T(n) = 3n+1 if n odd; trivial cycle 4 → 2 → 1 → 4; branching number c=4; automorphism group of the Cayley graph acts transitively on root paths; decidability of an+b conjectures reduces to graph automorphism properties. | CONNECTION: The branching number c=4 and the cyclic root 4→2→1→4 form a 4-cycle — a square, the simplest crystallographic lattice in 2D (square lattice symmetry p4m). The ratio of the cycle's edge lengths (if mapped to a geometric square) is 1:1, but the *walk lengths* to the root from any n scale logarithmically — the inverse of the golden ratio appears in the density of odd numbers in the Collatz sequence: for a random odd n, the expected number of divisions by 2 before next odd is $2$, giving a contraction factor of 3/4 per o Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -