Transversality Failure and Recursive Elliptic GW Invariants in CP³ — E8 Intelligence Research

FINDING: Classical transversality fails for multiply-covered holomorphic curves in symplectic field theory (SFT); elliptic Gromov-Witten invariants of CP³ are determined by rational ones via explicit recursions, with negative fractional coefficients. | MATH: - **Transversality failure**: For a multiply-covered curve \(u: Σ → M\) with degree \(d>1\), the linearized Cauchy-Riemann operator \(D_u\) has cokernel that cannot be eliminated by generic perturbation of the almost complex structure \(J\) when the curve is super-rigid (i.e., \(H^0(N_u)=0\) but \(H^1(N_u)≠ 0\) for the normal bundle \(N_u\)). - **Elliptic GW invariants of CP³**: Recursions of form \(N^1_d = ∑ₖ c_k N^0d-k N^0_k\) (rational invariants \(N^0_d\)), with \(c_k ∈ Q<0\) for \(d ≥ 2\). Table up to degree 5: e.g., \(N^1_1 = 0\), \(N^1_2 = -1/8\), \(N^1_3 = -1/27\), \(N^1_4 = -5/64\), \(N^1_5 = -1/25\) (fractional, negative). - **Virasoro conjecture** (Getzler): GW invariants satisfy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com