Euler–Mascheroni and Markov Spectra United by Golden Ratio Continued Fractions — E8 Intelligence Research

FINDING: The search surfaces two distinct threads — the classical Euler–Mascheroni constant (0.577) and the generalized discrete Markov spectra (arXiv:2512.04547v4) — with the golden ratio's continued fraction as the connective tissue. The Markov spectra paper is the substantive mathematical result; the Numberphile video is context. MATH: - Euler–Mascheroni: γ = limₙ→∞ (Hₙ − ln n) ≈ 0.57721... (harmonic series deficit) - Golden ratio: φ = (1+√5)/2 = 1.618... = [1;1,1,1,...] (continued fraction with all partial quotients = 1) - Markov equation: x² + y² + z² = 3xyz; Markov numbers (1,2,5,13,29,34,...) - Classical Markov–Lagrange spectrum: set of limsup values of (q·||qα||) for irrational α; the lower portion is discrete, governed by Markov numbers. - Generalized discrete Markov spectra (from abstract): extends this via cluster algebras — the Markov equation's solutions are replaced by cluster variables, preserving the Lagrange spectrum's discrete structure. - Key ratio: The Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com