Viswanath's Constant: The Almost-Sure Growth Rate of Random Fibonacci Sequences — E8 Intelligence Research

FINDING: Viswanath's constant (≈1.13198824) emerges as the almost-sure growth rate of random Fibonacci sequences, derived via Lyapunov exponent of a random matrix product. | MATH: Random Fibonacci recurrence: \( x_{n+1} = x_n \pm x_{n-1} \) with equal probability. Lyapunov exponent \( \lambda = \lim_{n\to\infty} \frac{1}{n} \log |x_n| \). Viswanath (1999) proved \( \lambda = \int_0^\infty \log t \, d\nu(t) \) where \( \nu \) is a specific invariant measure on continued fraction expansions, yielding \( \lambda \approx 0.123975598 \). Then growth constant \( C = e^\lambda \approx 1.13198824 \). | CONNECTION: No direct geometric harmony ratios (0.618, 1.618, etc.) appear. However, the invariant measure \( \nu \) is supported on a fractal set related to the Stern-Brocot tree and Farey sequences, which are intimately connected to the golden ratio and continued fractions. The golden ratio \( \phi = 1.618... \) appears as the growth rate for the deterministic Fibonacci sequence; Viswanath's c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com