PhysicsFormalization 🚀

A formal verification library for fundamental physics written in Lean 4 and supported by Mathlib 4[cite: 6].

This project aims to formally define and prove core physical theorems, bridging the gap between theoretical models, experimental bounds (such as relativistic PPN metrics), and quantum mechanical frameworks[cite: 6].

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📋 Overview

The workspace covers three main physical domains[cite: 6]:

1. Analytical Mechanics & Variational Principles[cite: 6]

   • Formalization of configuration paths, variations, and action functionals[cite: 6].
   • Rigorous formal statement of the Calculus of Variations and the Euler-Lagrange equations[cite: 6].
   • Modular definitions split across dedicated analytical mechanics submodules.

2. General Relativity & Post-Newtonian Formalism (PPN)[cite: 6]

   • Formalization of the Parameterized Post-Newtonian (PPN) parameter space ($\gamma, \beta, \alpha_1, \alpha_2$)[cite: 6].
   • Proof of Local Lorentz Invariance (LLI) under General Relativity ($\alpha_1 = 0$)[cite: 4, 6].
   • Orbital dynamics modeling anomalous node precession for satellite orbits ($\text{LAGEOS}$)[cite: 1, 2, 6].
   • Experimental verification bounds based on the SaToR-G experiment (INAF/INFN), establishing strict constraint checks ($\vert{}\alpha_1\vert{} \le 2 \times 10^{-5}$)[cite: 6].

3. Quantum Mechanics & Hilbert Spaces[cite: 6]

   • Definitions of normalized quantum states in complex Hilbert spaces ($\Vert{}\psi\Vert{} = 1$)[cite: 6].
   • Self-adjoint linear operators modeling physical observables ($\langle Ax, y \rangle = \langle x, Ay \rangle$)[cite: 6].
   • Formal proof of the Reality Theorem of Expectation Values: $\text{Im}(\langle \psi, A\psi \rangle) = 0$[cite: 6].

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🗂️ Project Structure

Based on the core library imports, the workspace is organized as follows:

PhysicsFormalization/
├── Physics/
│   ├── AnalyticalMechanics/
│   │   ├── Variation.lean        # Path variations and boundary conditions
│   │   ├── Action.lean           # Action functional and Lagrangian integration
│   │   └── EulerLagrange.lean    # Stationary action and Euler-Lagrange equations
│   ├── GeneralRelativity/
│   │   ├── PPN/
│   │   │   └── Basic.lean        # PPN parameters definition and GR LLI proof
│   │   ├── Dynamics/
│   │   │   └── Orbital.lean      # Orbital elements and LLI-induced precession
│   │   └── Experiments/
│   │       └── SatorG.lean       # Experimental bounds and compatibility proofs
│   └── QuantumMechanics/
│       └── Basic.lean            # Quantum states, observables, and expectation values
├── Physics.lean                  # Library entry point and module exports
├── lakefile.lean                 # Lake build configuration
└── lean-toolchain                # Lean 4 toolchain specification

⚙️ Environment & Requirements
To successfully build and verify this library, ensure you have the following toolchain versions installed:
Lean 4 Version: v4.33.0-rc1 (or compatible versions specified in lean-toolchain)
Dependency Manager: elan (the Lean version manager) and lake (Lean's build system)
You can manage Lean toolchains easily via elan.

🛠️ Build & Compilation Instructions
Clone the repository:
Bash
git clone [https://github.com/toshiro881967-coder/PhysicsFormalization.git](https://github.com/toshiro881967-coder/PhysicsFormalization.git)
cd PhysicsFormalization
Fetch dependencies and build Mathlib/Project components:
lake will automatically download the correct Lean toolchain specified in the lean-toolchain file and fetch precompiled Mathlib build caches.
Bash
lake update
lake build
Verify individual modules:
To check a specific file (e.g., the variational mechanics module):
Bash
lake build Physics.AnalyticalMechanics.EulerLagrange