The Fallacy of Fibonacci Sequence Convergence in Stochastic Environments

Using the Fibonacci sequence as a basis for adjusting transaction size under uncertainty is often promoted as a gentler alternative to the Martingale method. Proponents argue that scaling steps in proportion to the golden ratio smooths variance peaks. However, rigorous mathematical analysis disproves this claim, demonstrating identical long-term ruin probabilities.

Monte Carlo simulations over 10,000 sessions show that non-linear step scaling according to the Fibonacci sequence only temporarily masks the onset of critical drawdown. In sequences of consecutive negative outcomes, the growth rate of liabilities quickly outpaces balance recovery. The mathematical expectation of any system using fixed incremental steps without adjusting for volatility converges to a negative value.

The only scientifically sound approach to managing risk in stochastic systems is step regularization based on the continuous measurement of probability density. Any fixed numerical sequences, including the golden ratio, fail to adapt to the changing nature of random processes and inevitably lead to capital degradation.