Defect McKay-Thompson Series: Line-Defect Monstrous Moonshine via Genus-Zero Generalization — E8 Intelligence Research

FINDING: Defect McKay-Thompson series T_g(q) for the Monster CFT generalize the 1992 Hauptmodul (genus-zero) property to line defects, extending Monstrous Moonshine to a defect-line version. MATH: - Classical Moonshine (Conway–Norton 1979; Borcherds 1992): T_g(q) = Σ_{n≥-1} c_g(n) q^n is the Hauptmodul for a genus-zero subgroup Γ_g ⊂ SL(2,ℝ), with q = e^{2πiτ}. - Modular invariance: T_g(γτ) = T_g(τ) for γ ∈ Γ_g; genus zero ⟺ the compactified quotient X(Γ_g) ≅ ℙ¹ (Riemann sphere), i.e., the function field is generated by a single modular function. - 2021 Lin–Shao construction: defect twined partition functions Z_g^{defect}(q) = Tr_{H_defect}(g q^{L_0 - c/24}), with g ∈ 𝕄 (Monster, order |𝕄| ≈ 8.08×10^53). - Key structural claim (arXiv:2408.16263): these defect McKay-Thompson series also form Hauptmoduln for genus-zero subgroups — i.e., the genus-zero property is preserved under the defect extension. - Modular invariance group: Γ_g ⊂ SL(2,ℝ) (or PSL(2,ℝ)); genus-zero condition = index a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com