Genus-Zero Hauptmoduln as the Classification Criterion for Moonshine — E8 Intelligence Research

FINDING: The McKay–Thompson series T_g(q) of the Moonshine module are Hauptmoduln (principal moduli) for genus-zero subgroups of SL(2,ℝ), and this genus-zero property is the structural criterion that classifies which "moonshine" correspondences exist — extended to umbral moonshine (labelled by Niemeier lattices / ADE root systems) and to defect/twined partition functions. MATH: - j-function: j(τ) = q⁻¹ + 744 + 196884q + 21493760q² + …, q = e^{2πiτ}. Coefficient 196884 = 196883 + 1 = (dim of smallest nontrivial Monster irrep) + 1. - McKay–Thompson series: T_g(q) = Σ_{n} tr(g | V_n) q^{n−1}, with V = V^♮ the Moonshine VOA (central charge c = 24). - Genus-zero condition: the compactified quotient X_0(g) = ℍ*/Γ_g has genus 0, so its function field is generated by a single Hauptmodul; T_g is that Hauptmodul. - Modularity: T_g(τ) is a modular function for Γ_g ⊂ SL(2,ℝ) commensurable with SL(2,ℤ), invariant under Γ_g, with integer Fourier coefficients. - Umbral moonshine: labelled by Niemeie Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com