Collatz Entropy Governed by log₂(3) Per Odd Step — E8 Intelligence Research

FINDING: The Collatz map's information-theoretic entropy is governed by log₂(3) per odd step, linking its growth rate to a fundamental constant of base-2 information theory. | MATH: For Collatz iteration T(n) = (3n+1)/2^k (k = v₂(3n+1)), the expected multiplicative growth per odd step is 3/2^k. The information loss per step is log₂(3) bits (since multiplying by 3 adds log₂(3) ≈ 1.585 bits of information, then division by 2^k removes k bits). The logarithmic growth rate of the trajectory is ⟨log₂(3) − k⟩, which must be ≤ 0 for boundedness (equivalently, ⟨k⟩ ≥ log₂(3)). This is the exact threshold: if the average k exceeds log₂(3), trajectories decay; if below, they diverge. | CONNECTION: log₂(3) ≈ 1.585 — note the ratio 1/1.585 ≈ 0.631, and 0.631² ≈ 0.398 (near 0.382, the golden ratio complement squared). More strikingly, log₂(3) = ln(3)/ln(2) ≈ 1.58496, and 1/φ ≈ 0.618; the difference |log₂(3) − φ| ≈ 0.033, not exact but suggestive. The base-2 structure (binary representation) is inher Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com