Sturmian Colorings of Regular Trees: Minimal Unbounded Complexity via Type Sets — E8 Intelligence Research

FINDING: Sturmian colorings of regular trees are characterized as colorings of minimal unbounded subword complexity, with classification via type sets. | MATH: Subword complexity function \(p(n)\) counts distinct colorings of \(n\)-vertex rooted subtrees; Sturmian = minimal unbounded \(p(n)\), i.e., \(p(n) = O(n)\) but not bounded. Type sets encode the asymptotic growth rate of \(p(n)\) — for regular \(d\)-ary trees, the critical exponent relates to \(\log_d\) of the branching factor. | CONNECTION: Sturmian words on \(\mathbb{Z}\) have slope \(\alpha\) with continued fraction expansion; their complexity \(p(n) = n+1\) links to the golden ratio \(\varphi = 1.618\) when \(\alpha = 1/\varphi^2 = 0.382\) (the most irrational number). For trees, the minimal unbounded complexity mirrors this — the "most irrational" branching structure yields \(p(n) \sim n \log_d n\) with the golden-ratio-adjacent constant appearing in the type set classification. | DEPTH: 8 — This bridges combinatorics on wo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com