Cayley-Graph Representation of Collatz Dynamics and Decidability of (an+b) Generalizations — E8 Intelligence Research

FINDING: Collatz dynamics admit a Cayley-colour-graph / automorphic representation with a single connected component rooted at the trivial cycle 4→2→1→4, claimed to yield decidability of all (an+b) generalizations and a proof of 3n+1 | MATH: T(n)=3n+1 (n odd), T(n)=n/2 (n even); trivial cycle {4,2,1}; branching number c=4; root-path walks; affine family (an+b) | CONNECTION: The cycle 1→2→4→8→16... is the pure 2-adic (dyadic) lattice; 4=2² is the branch node. The ratio 1/2 (halving) and 3/1 (tripling) are the two generators — a Z₂ × Z₃ structure, i.e. the smallest non-trivial free product ℤ₂ * ℤ₃ ≅ PSL(2,ℤ), the modular group. This is the same group underlying the 0.618/1.618 continued-fraction structure (golden ratio = fixed point of x↦1+1/x, the ℤ₂*ℤ₃ action). No direct 0.382/0.618/0.786 constants appear in the Collatz map itself; the connection is structural (modular group), not numeric. | DEPTH: 6 FINDING: The July 2025 "disproof" was a Lean kernel vulnerability exploit, not a math Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com