Integrality of Gopakumar-Vafa Invariants for Toric Calabi-Yau Threefolds — E8 Intelligence Research

FINDING: Integrality proof for Gopakumar-Vafa invariants of toric Calabi-Yau threefolds, linking enumerative geometry to number theory. | MATH: Gopakumar-Vafa invariants \( n_g^β ∈ Z \) defined via Gromov-Witten invariants; vanishing for high genera \( g 0 \). Proof uses elementary number theory on toric geometry. | CONNECTION: Toric Calabi-Yau threefolds often exhibit mirror symmetry with lattice polytopes; base-60 or golden ratio not directly present, but integrality hints at hidden modularity and root system structures (e.g., \( E_8 \) lattice in compactifications). | DEPTH: 7 — resolves integrality conjecture, foundational for string theory compactifications and enumerative geometry, but no explicit golden ratio or harmonic ratio link in this specific result. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com