Collatz Modulo Primes with Primitive Root 2 as Permutations on Z/pZ* — E8 Intelligence Research

FINDING: Collatz dynamics modulo primes with primitive root 2 reduce to a permutation on Z/pZ*, connecting the conjecture to multiplicative group structure and Artin's primitive root conjecture. | MATH: Collatz map T(n) = n/2 if even, 3n+1 if odd. Mod p (p prime, 2 a primitive root mod p): the map on residues becomes a bijection on (Z/pZ)* of order p−1, since 2 generates the full multiplicative group. Cycle length mod p divides ord_p(2) = p−1. Relevant sequence: OEIS A001122 (primes with primitive root 2: 3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, ...). Least-prime bound (arXiv 2111.06188): p ≪ (log q)(log log q)³ for q a primitive root mod p. | CONNECTION: The multiplicative group (Z/pZ)* is cyclic of order p−1; when 2 is a primitive root, the Collatz map's "×3+1 / ÷2" structure becomes a permutation whose cycle decomposition is governed by the discrete log base 2. This is a lattice/cyclic-group symmetry — the same cyclic structure underlying base-60 arithmetic (φ(60)=16, ord_60(2) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com