Collatz Cycles and Prisoner Permutations: Hidden Symmetry in Integer Dynamics — E8 Intelligence Research

Abstract

FINDING: The Collatz Conjecture — a deceptively simple iterative map — remains unproven, encoding hidden structure in integer dynamics; the 100 Prisoners Riddle reveals a cycle-decomposition symmetry tied to permutation statistics. | MATH: Collatz map: \( T(n) = \begin{cases} n/2 & \text{if } n \equiv 0 \pmod{2} \\ 3n+1 & \text{if } n \equiv 1 \pmod{2} \end{cases} \). No closed-form invariant known; empirical stopping times show fractal-like scaling. 100 Prisoners: optimal strategy uses permutation cycles — success probability = \( \sum_{k=1}^{n} \frac{1}{k} - \sum_{k=1}^{n} \frac{1}{k+1} \approx 1 - \ln 2 \approx 0.30685 \) for \( n \to \infty \). | CONNECTION: Collatz stopping-time distributions exhibit self-similarity reminiscent of logarithmic spirals (ratio ~1.618 appears in empirical scaling of max excursion vs. input size in some statistical fits, though not rigorously proven). The 100 Prisoners solution relies on cycle structure — a direct analogue to permutation groups and roo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com