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 \(\alpha\) (irrational), the \(\ell\)-boundary sequence \(B_\ell(n)\) is the finite set of pairs \((u,v)\) of prefixes/suffixes of length \(\ell\) in factors \(uyv\) of length \(n+\ell\). The paper generalizes Chen–Wen boundary sequences. Key structural result: for Sturmian words, \(B_\ell(n)\) is periodic in \(n\) with period related to the continued fraction expansion of \(\alpha\). Specifically, the number of distinct boundary pairs is bounded by \(\ell+1\), and the sequence is ultimately periodic with period \(q_k\) where \(q_k\) is the denominator of the \(k\)-th convergent of \(\alpha\). For the golden ratio \(\alpha = (\sqrt{5}-1)/2 = 1/\varphi \approx 0.618\), 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