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

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: \( T(n) = {cases} n/2 & if n ≡ 0 {2} \\ 3n+1 & if n ≡ 1 {2} {cases} \). No closed-form invariant known; empirical stopping times show fractal-like scaling. 100 Prisoners: optimal strategy uses permutation cycles — success probability = \( ∑ₖ₌₁ⁿ 1/k - ∑ₖ₌₁ⁿ 1/k+1 ≈ 1 - ln 2 ≈ 0.30685 \) for \( n → ∞ \). | 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