Sturmian Repetition Invariance and Self-Similarity via Irrational Slopes — E8 Intelligence Research

Abstract

FINDING: Sturmian words — minimal-complexity infinite binary sequences — are fully characterized by irrational slopes; their repetition exponent is invariant under suffixing, and for quadratic-irrational slopes the structure locks into self-similar continued-fraction patterns. | MATH: Sturmian word complexity \(p(n) = n+1\) (minimal aperiodic); slope \(\theta \in \mathbb{R}\setminus\mathbb{Q}\); characteristic Sturmian word \(c_\theta\) has repetition exponent \(\mathrm{rep}(c_\theta) = \mathrm{rep}(y)\) for every suffix \(y\); for quadratic irrational \(\theta\), \(\mathrm{rep}(c_\theta)\) relates to the continued fraction of \(\theta\) (e.g., \(\theta = [0; \overline{a_1,\dots,a_k}]\) yields \(\mathrm{rep} = 2 + \sup\) of partial quotient sums). | CONNECTION: Quadratic irrational slopes \(\theta\) have continued fractions with periodic partial quotients — these periods generate self-similar scaling. For \(\theta = (\sqrt{5}-1)/2 \approx 0.618\), the golden ratio conjugate, the contin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com