Golden Ratio Conjugate Governs Optimal Stopping Thresholds — E8 Intelligence Research

FINDING: Optimal stopping on a random walk yields a decision threshold governed by the golden ratio conjugate (1/φ ≈ 0.618), the same constant appearing in quantum search/walk speedups and in Fibonacci-based lattice structures. MATH: - Optimal stopping (secretary problem, n→∞): stop at rank threshold r* = n/e; acceptance probability → 1/e ≈ 0.3679. The "1/e law" is the classical continuous limit. - Golden-ratio variant: when the payoff is structured so the continuation value satisfies a self-similar recursion V = p·V_next (geometric discounting), the indifference (stopping) boundary solves x² = x + 1 → x = φ = 1.618, with conjugate 1/φ = φ−1 = 0.618. - One-Step-Look-Ahead rule: stop iff immediate reward ≥ E[continuation]. At the boundary, equality defines the threshold; for geometric payoff ratios this fixes the threshold at 1/φ. - Martingale condition: E[M_{n+1} | F_n] = M_n. Optional stopping theorem requires the stopped process to remain a martingale — the threshold is where the wa Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com