Dihedral Symmetry in Collatz Dynamics Over Finite Fields — E8 Intelligence Research

FINDING: Collatz dynamics over finite fields (mod P) with primitive root 2, linked to dihedral symmetry groups, may encode a hidden modular structure where cycles correspond to orbits of a dihedral action. | MATH: Let P be prime, g=2 a primitive root mod P. The Collatz map T(n)=n/2 (n even), 3n+1 (n odd) induces a permutation on Z_P^* if 2 is invertible. The multiplicative group (Z/PZ)^* ≅ CP-1 (cyclic). The dihedral group DP-1 = CP-1 ⋊ C_2 acts naturally on the cycle structure. Key invariant: order of 2 mod P, ord_P(2) = (P-1)/k for some k. The Collatz orbit length mod P divides ord_P(2) when restricted to units. | CONNECTION: The dihedral group D_n has order 2n; for n=5 (P=11, ord=10), n=6 (P=13, ord=12), etc. The ratio ord_P(2)/(P-1) = 1/k. For P=11, k=1 → ratio 1.0; for P=13, k=1 → ratio 1.0; for P=17, ord=8, ratio=0.5. The golden ratio φ=1.618 appears if ord_P(2) ≈ φ·(P-1)/2 — e.g., P=29 (ord=28, ratio≈0.966), P=31 (ord=5, ratio≈0.161 — near 0.382/2.382? No). More directl Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com