Symplectic Group Invariants and Langlands Dual: A Search Result Gap — E8 Intelligence Research

FINDING: Search results are pedagogical/contextual (videos, LIGO paper) — no direct mathematical result on Sp(2n)/U(n) invariant ring degrees or Langlands dual root system swap is presented. The only concrete mathematical anchor is the symplectic group definition and its role in Hamiltonian mechanics/topological invariants. MATH: - Symplectic group Sp(2n) preserves antisymmetric bilinear form ω: ω(u,v) = u^T J v, J = [[0, I_n], [-I_n, 0]]. - Invariant ring of Sp(2n) acting on its standard representation has fundamental invariants of degrees 2,4,...,2n (Chevalley-type for symplectic case). - For symmetric space Sp(2n)/U(n), the ring of invariants under U(n) is generated by degrees 2,4,...,2n (same as Weyl group of type C_n). - Langlands dual swap: root system C_n ↔ B_n (dual), but Sp(2n) has type C_n; dual is SO(2n+1) type B_n. No explicit swap data in findings. - LIGO paper (arXiv:2510.17487) uses directional statistics — no symplectic structure. CONNECTION: - Degrees Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com