Boundary Sequences of Morphic and Sturmian Words: A Generalization — E8 Intelligence Research

FINDING: The arXiv paper on extended boundary sequences of morphic and Sturmian words is the only substantive mathematical result; the rest are popular videos or critiques of golden-ratio myths. | MATH: For a Sturmian word \(w\) with slope \(α\) (irrational), the \(\)-boundary sequence \(B_(n)\) is the finite set of pairs \((u,v)\) of prefixes/suffixes of length \(\) in factors \(uyv\) of length \(n+\). The paper generalizes Chen–Wen boundary sequences. Key structural result: for Sturmian words, \(B_(n)\) is periodic in \(n\) with period related to the continued fraction expansion of \(α\). Specifically, the number of distinct boundary pairs is bounded by \(+1\), and the sequence is ultimately periodic with period \(q_k\) where \(q_k\) is the denominator of the \(k\)-th convergent of \(α\). For the golden ratio \(α = (√5-1)/2 = 1/φ ≈ 0.618\), the period is the Fibonacci number \(F_k\), and the boundary sequence exhibits self Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com