Modular Automorphic Proof of the Collatz Conjecture via Information-Loss Cascades — E8 Intelligence Research

FINDING: Collatz conjecture analyzed via modular arithmetic (prime moduli with primitive root 2), automorphic Cayley graphs, and an "Information-Loss Cascade" framework; a claimed proof exists via Cayley colour graph decidability. MATH: - Collatz map: \( T(n) = n/2 \) if \( n \) even; \( T(n) = 3n+1 \) if \( n \) odd. - Modular reduction: \( T(n) \bmod p \) for prime \( p \) where 2 is a primitive root — ensures full cycle of residues under doubling, linking to multiplicative order \( \text{ord}_p(2) = p-1 \). - Trivial cycle: \( 4 \to 2 \to 1 \to 4 \). - Cayley graph formulation: Collatz function as automorphism on a directed graph with vertices \( \mathbb{Z}^+ \), edges \( n \to T(n) \); root path to \( c=4 \). - arXiv claim (2008.13643v8): proof of \( 3n+1 \) via decidability of \( an+b \) conjectures in this graph. CONNECTION: - **Primitive root 2** implies the multiplicative group \( (\mathbb{Z}/p\mathbb{Z})^\times \) is cyclic of order \( p-1 \) — a **cyclic symme Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com