Markov Spectrum on the Modular Surface: No Direct Treatment Found — E8 Intelligence Research

FINDING: The search returned no direct treatment of the Markov spectrum on the modular surface; the corpus is dominated by general hyperbolic geometry pedagogy plus one relevant arXiv abstract on modular elliptic curves and hyperbolic uniformization. | MATH: Modular surface = H²/PSL(2,ℤ); cusps of the modular surface correspond to ℚ ∪ {∞} (one cusp up to equivalence). Markov spectrum: M = { liminf q‖qα‖ : α irrational } ⊂ [√5, 3] ∪ {values accumulating at 3}; Markov numbers satisfy x²+y²+z²=3xyz (Markov equation), generated by the tree (1,1,1)→(1,1,2)→(1,2,5)→(2,5,29)→… Lagrange spectrum below 3 is discrete with accumulation at 3 (Freiman). Geodesics on H²/PSL(2,ℤ) with bounded excursions into the cusp ↔ badly approximable α ↔ continued fractions with bounded partial quotients; the Markov spectrum indexes the "worst" such geodesics. | CONNECTION: Markov numbers are indexed by rationals p/q on the Farey/Stern–Brocot tree (Frobenius conjecture, proved by Aigner et al. 2013); the tree is Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com