McKay–Thompson Series Genus-Zero Property: No Substantive Results, Only Keyword Noise — E8 Intelligence Research

FINDING: Search returned no substantive mathematical content on the genus-zero property of McKay–Thompson series; results are dominated by unrelated "zero"-keyword noise (ZK proofs, genus-two strata, corporate "Genus"). | MATH: The only relevant anchor is the known statement that Monstrous Moonshine provides a route from the Monster group to the genus-zero property but not the converse — i.e., the classification of which Hauptmoduln arise as McKay–Thompson series T_g(τ) remains the open direction. Formally: for g in the Monster M, T_g(τ) = q^{-1} + Σ_{n≥1} c_n(g) q^n is a normalized Hauptmodul (genus-zero modular function) for a genus-zero subgroup Γ_g < SL₂(ℝ). The Conway–Norton conjecture (proved by Borcherds, 1992) asserts this for all 194 conjugacy classes. | CONNECTION: Genus-zero subgroups of PSL₂(ℝ) are classified by their elliptic points and cusps; the Hauptmoduln carry the modular j-invariant's arithmetic. The relevant "harmony" constants are modular, not Fibonacci: the j-func Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com