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

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 \(θ ∈ R\); characteristic Sturmian word \(c_θ\) has repetition exponent \(rep(c_θ) = rep(y)\) for every suffix \(y\); for quadratic irrational \(θ\), \(rep(c_θ)\) relates to the continued fraction of \(θ\) (e.g., \(θ = [0; a_1,,a_k̄]\) yields \(rep = 2 + \) of partial quotient sums). | CONNECTION: Quadratic irrational slopes \(θ\) have continued fractions with periodic partial quotients — these periods generate self-similar scaling. For \(θ = (√5-1)/2 ≈ 0.618\), the golden ratio conjugate, the contin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com