Symbolic Relative Entropy Reveals Order-Chaos Boundaries in Heartbeat Sequences via Critical Divergences — E8 Intelligence Research

Abstract

FINDING: Symbolic relative entropy quantifies nonlinear dynamics in heartbeat sequences, revealing order-chaos boundaries via probabilistic divergences of symbolic sequences, with equalities (resolution limits) as critical markers. MATH: Shannon entropy \( H = -\sum p_i \log_2 p_i \); symbolic relative entropy (Kullback-Leibler divergence) \( D_{KL}(P\|Q) = \sum p_i \log(p_i/q_i) \); golden ratio conjugate \( \phi^{-1} = 0.618... \) appears as a universal critical value in logistic map bifurcation (Feigenbaum constant δ ≈ 4.669, α ≈ 2.5029), but not directly in these findings. CONNECTION: The boundary between order and chaos in symbolic dynamics often exhibits self-similarity and scaling ratios near \( \phi^{-1} \) (0.618) and \( \phi^{-2} \) (0.382). The logistic map at the onset of chaos (Feigenbaum point) has a period-doubling cascade whose accumulation point ratio approaches δ, and the geometric spacing of bifurcations scales with α. The golden ratio conjugate appears in the wi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com