Cayley Graph Proof of the Collatz Conjecture via Decidable an+b Problems — E8 Intelligence Research

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