Moncrief's Reduced Phase Space from Symplectic Reduction of the Corner Symmetry Group

We show that the cotangent bundle of Teichmüller space T𝒯(Σ_g) — Moncrief's reduced phase space for (2+1) vacuum gravity — arises by symplectic reduction of the zero-enstrophy coadjoint orbit of the corner symmetry group Diff(S) ⋉ SL(2,ℝ)^S by SDiff(S). The zero-enstrophy condition (vanishing normal bundle curvature K⊥) decouples the codimension-2 Codazzi equations, producing holomorphic quadratic differentials as the cotangent fiber. The Ricci equation forces the two normal-direction differentials to be real-proportional at all genera, by the open mapping theorem. The symplectic structure is verified via the Chern-Simons triangle: the corner symplectic potential, restricted to flat normal connections and reduced by SDiff, reproduces the canonical symplectic form on T𝒯 up to the gravitational normalisation 1/(8πG). This connects the canonical reduction programme for (2+1) gravity with the coadjoint orbit classification of gravitational edge modes, identifying Moncrief's phase space as a distinguished orbit with maximally large little group.Version 2: Revised for journal submission. This version sharpens the theorem/conjecture boundary, adds the explicit nonlinear torus construction and symplectic normalization, expands the Lean 4 verification statement, and includes complete data, software, funding, conflict-of-interest, and generative-AI disclosures.