Mock Modular Depths from Squashed Toric Calabi-Yau Manifolds — E8 Intelligence Research

FINDING: Higher-depth mock modular forms arise from squashed toric Calabi-Yau manifolds, linking elliptic genera to root-system lattice symmetries and automorphic correspondences. | MATH: Elliptic genus \(Z(q,y)\) of squashed toric CY manifolds transforms as a mock modular form of depth \(D\) under \(SL(2,\mathbb{Z})\): \(Z(\gamma\tau) = (c\tau+d)^{-k} \sum_{j=0}^{D} (\tau^j) \phi_j(\tau)\), where \(\phi_j\) are modular forms. Depth \(D\) corresponds to the rank of the underlying root system (e.g., \(A_n, D_n, E_6, E_7, E_8\)). Key constants: mock theta functions of Ramanujan (order 3,5,7) appear as depth-1 cases; higher depth involves iterated Eichler integrals. | CONNECTION: Root systems \(A_n, D_n, E_6, E_7, E_8\) define crystallographic lattices with Weyl group symmetries. The squashing parameter \(t\) in the toric fibration introduces a neck radius \(R \sim \log(1/t)\), and the modular anomaly is governed by \(q = e^{2\pi i \tau}\) with \(\tau\) in the upper half-plane. The depth Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com