SL2RGeometry.lean

/-
Copyright (c) 2026. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antigravity
-/
import Formalization.SL2RGeometry.Basic
import Formalization.SL2RGeometry.Geometry
import Formalization.SL2RGeometry.SpectralDecomposition

/-!
# Pillar 6: $\widetilde{\mathrm{SL}}_2(\mathbb{R})$ Geometry — Lie Structure, Riemannian Metric & Casimir Spectrum

This master module aggregates the complete Lean 4 formalization suite for **Pillar 6: $\widetilde{\mathrm{SL}}_2(\mathbb{R})$ Geometry**,
one of Thurston's eight model 3-dimensional geometries, realizing compact 3-manifold quotients as unit tangent bundles
$T^1(\Sigma_g) \cong \widetilde{\Gamma}_g \backslash \widetilde{\mathrm{SL}}_2(\mathbb{R})$ over closed hyperbolic surfaces
of genus $g \ge 2$.

## Module Index & Architecture

1. **`Formalization.SL2RGeometry.Basic`**:
   - **$\mathfrak{sl}_2(\mathbb{R})$ Lie Algebra Basis & Matrix Representation**:
     Basis $e_1 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, e_2 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, e_3 = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$.
     Commutation relations: $[e_1, e_2] = 2e_3, [e_2, e_3] = -2e_1, [e_3, e_1] = -2e_2$.
     Tracelessness $\operatorname{Tr}(e_i) = 0$ and Jacobi identity.
   - **Universal Cover $\widetilde{\mathrm{SL}}_2(\mathbb{R})$ & Central Extension**:
     Infinite cyclic center $Z(\widetilde{\mathrm{SL}}_2(\mathbb{R})) \cong \mathbb{Z}$ and central extension
     $1 \to \mathbb{Z} \to \widetilde{\mathrm{SL}}_2(\mathbb{R}) \to \mathrm{PSL}_2(\mathbb{R}) \to 1$.
   - **Unit Tangent Bundle $T^1(\Sigma_g)$ Topology ($g \ge 2$)**:
     Euler characteristic $\chi(\Sigma_g) = 2 - 2g < 0$.
     Circle bundle Euler class $e(T^1(\Sigma_g)) = \chi(\Sigma_g) = 2 - 2g < 0$ and $|e| = 2g - 2 > 0$.
     Surface area $\operatorname{Area}(\Sigma_g) = 4\pi(g - 1)$ and volume $\operatorname{Vol}(T^1(\Sigma_g)) = 4\pi^2(g - 1) > 0$.
   - **First Homology & Betti Numbers**:
     Abelianization $H_1(T^1(\Sigma_g), \mathbb{Z}) \cong \mathbb{Z}^{2g} \oplus \mathbb{Z}/(2g - 2)\mathbb{Z}$.
     First Betti number $b_1(T^1(\Sigma_g)) = 2g$.
     Torsion order $|H_1^{\mathrm{tor}}| = 2g - 2 = -\chi(\Sigma_g) > 0$.

2. **`Formalization.SL2RGeometry.Geometry`**:
   - **Riemannian Metric & Connection 1-Form**:
     Left-invariant orthonormal frame $(E_1, E_2, E_3)$ and connection 1-form $\omega = \omega^3$ dual to $E_3$.
     Standard metric $g = (\omega^1)^2 + (\omega^2)^2 + (\omega^3)^2$.
   - **Sectional Curvatures**:
     Horizontal curvature $K(E_1, E_2) = -3/4 < 0$.
     Vertical-horizontal curvatures $K(E_1, E_3) = +1/4 > 0, K(E_2, E_3) = +1/4 > 0$.
     Mixed curvature bounds $\min K = -3/4 < 0 < +1/4 = \max K$.
   - **Ricci Curvature Tensor**:
     $R_{11} = -1/2, R_{22} = -1/2, R_{33} = +1/2$ with vanishing off-diagonal components $R_{12} = R_{13} = R_{23} = 0$.
   - **Scalar Curvature & Ricci Anisotropy**:
     Constant negative scalar curvature $R = R_{11} + R_{22} + R_{33} = -1/2$.
     Ricci anisotropy ratio $R_{33} / R_{11} = -1$.

3. **`Formalization.SL2RGeometry.SpectralDecomposition`**:
   - **Fiberwise Fourier Decomposition**:
     $L^2(T^1(\Sigma_g)) = \bigoplus_{m \in \mathbb{Z}} \mathcal{H}_m$ with $f = \sum_{m \in \mathbb{Z}} f_m e^{i m \theta}$.
   - **Casimir & Laplace-Beltrami Eigenvalues**:
     $\lambda_{j, m} = \lambda_j(\Sigma_g) + \frac{m^2}{4}$.
   - **Base Selberg Spectrum Connection**:
     For $m = 0$, $\lambda_{j, 0} = \lambda_j(\Sigma_g)$.
     Ground state $\lambda_{0, 0} = 0$, vertical ground state $\lambda_{0, 1} = 1/4$.
   - **Positive Spectral Gap**:
     $\lambda_1(T^1(\Sigma_g)) = \min(\lambda_1(\Sigma_g), 1/4) > 0$.
     For all non-trivial states $(j, m) \ne (0, 0)$, $\lambda_{j, m} > 0$.
     Monotonicity in base eigenvalue and vertical angular momentum magnitude $|m|$.

## Academic & Mathematical References

- **Milnor, J.** (1976). *Curvatures of left invariant metrics on Lie groups*.
  *Advances in Mathematics*, 21(3), 293–329.
- **Scott, P.** (1983). *The geometries of 3-manifolds*.
  *Bulletin of the London Mathematical Society*, 15(5), 401–487.
- **Thurston, W. P.** (1997). *Three-Dimensional Geometry and Topology* (Vol. 1).
  *Princeton University Press*, Edited by Silvio Levy.
- **Selberg, A.** (1956). *Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series*.
  *Journal of the Indian Mathematical Society*, 20, 47–87.
- **Buser, P.** (1992). *Geometry and Spectra of Compact Riemann Surfaces*.
  *Progress in Mathematics*, Birkhäuser Boston.
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