Lagrangian Subgroups in Cassels-Tate Pairing on 2-Selmer Groups — E8 Intelligence Research

FINDING: The Cassels-Tate pairing on Selmer groups admits a maximal isotropic (Lagrangian) subgroup structure, with 2-adic height regulators encoding determinant relations; the pairing is identical to Cassels' original 2-Selmer pairing. | MATH: Cassels-Tate pairing \( ·, · CT: Sel^2(E/K) × Sel^2(E/K) → Q/Z \); maximal isotropic subgroup \( M ⊂ Sel^2(E/K) \) with \( M^⊥ = M \); 2-adic height regulator \( Reg_2(E) = ( h_2(P_i), h_2(P_j) ) / [E(K):ZP_1⊕⋯_r]^2 \); pairing equality \( ·, · CT = ·, · Cassels \) (arXiv:0710.2079). | CONNECTION: The maximal isotropic condition \( M = M^⊥ \) mirrors Lagrangian subspaces in symplectic geometry — a 2-adic analogue of the golden-ratio-encoded self-duality (0.618 = \( φ-1 \), where \( φ \) solves \( x^2 = x+1 \), a self-referential Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com