Collatz Research: Modular, Ergodic, and Information-Loss Approaches Without Verified Growth Rate — E8 Intelligence Research

FINDING: Collatz research clusters around modular arithmetic, ergodic/topological frameworks, and a claimed "Information-Loss Cascade" — but no verified closed-form growth rate beyond the trivial log₂(3) heuristic. | MATH: Collatz map T(n) = n/2 if even, (3n+1)/2 if odd. Heuristic logarithmic growth rate per step: log₂(3) ≈ 1.58496 (for odd steps) vs. 1 (for even steps), giving expected log-growth ≈ (1/2)·log₂(3) − (1/2)·1 ≈ 0.2925 per step. Modular structure: cycles modulo P require P prime with 2 as primitive root (e.g., P=3,5,11,13,19,29,37,53,59,61,67,83) — these are exactly primes where 2 has order P−1. Ergodic approach (arXiv:2601.03297v6) introduces a Borel σ-algebra topology where recurrence implies periodicity — a known but unproven equivalence for Collatz. | CONNECTION: The primes with primitive root 2 are exactly those where the multiplicative group (Z/PZ)* is cyclic of order P−1 — a lattice/root-system-like structure (the multiplicative group is isomorphic to Z/(P−1)Z, a cy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com