Near-Horizon Extremal Geometry: Central Charge 12J via Symplectic Phase Space — E8 Intelligence Research

FINDING: Near-horizon extremal geometries admit an sl(2,R) × u(1) symmetry algebra with central charge 12J, where J is the angular momentum; symplectic phase-space structure underlies this. MATH: - Symmetry algebra: **sl(2,R) ⊕ u(1)** (Virasoro subalgebra with central charge **c = 12J**). - Symplectic form: **ω = dq ∧ dp** (invariant under Hamiltonian flow; phase-space area preserved). - For extremal Kerr/Reissner-Nordström near-horizon limits, the isometry group is **SL(2,R) × U(1)** — the U(1) corresponds to azimuthal rotation, and the SL(2,R) acts on the (t, r) sector. - Central charge relation: **c_L = 12J** (left-moving), with right-moving **c_R = 12J** for extremal Kerr (from Kerr/CFT correspondence). - No explicit golden-ratio or base-60 constants appear in the provided abstracts; the key constant is **J** (angular momentum) and the integer **12** (from the central charge formula). CONNECTION: - **sl(2,R)** is the Lie algebra of **SL(2,R)**, whose maximal compa Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com