Fractional Laplacian on Curved Manifolds

Computational verification code for: "Fractional Laplacian on Curved Manifolds: Coordinate-Invariant Construction and Physical Applications"

Author: Oksana Sudoma Date: 15 November, 2025

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Overview

This repository provides computational validation for coordinate-invariant fractional Laplacian operators on curved manifolds, with rigorous eigenvalue analysis on the 2-sphere demonstrating sub-percent accuracy and exact dimensional scaling.

Key discoveries:

• Curvature correction accuracy: 0.59% error in weak regime (κ_α < 0.1)
• R^-2 dimensional scaling: Exact to machine precision (10^-15)
• Spectral eigenvalue validation: All 51 correction signs correct
• Weak curvature threshold: κ_α < 0.1 boundary empirically confirmed

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Experiments

E53: Sphere Validation (S²)

Validates coordinate-invariant construction on 2-sphere.

Key results:

• Coordinate invariance: 10^-10 precision
• Eigenvalue spectrum verified
• Spectral discretization validated
• Runtime: ~2 seconds

Quick start:

cd experiments/E53_sphere_validation
python3 -m venv venv
source venv/bin/activate
pip install -r requirements.txt
python3 main.py

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Repository Structure

Fractional-Laplacian/
├── experiments/
│   └── E53_sphere_validation/           # Sphere curvature validation
│       ├── src/                         # Core implementation modules
│       ├── tests/                       # Unit tests
│       ├── outputs/
│       │   ├── EXPERIMENTAL_REPORT_E53.md    # Results (execution day)
│       │   ├── ANALYTICAL_REPORT_E53.md      # Analysis (next day)
│       │   └── data/                         # Validated numerical results
│       ├── ARCHITECTURE.md              # Implementation architecture
│       ├── README.md                    # Experiment pre-registration
│       ├── main.py                      # Entry point
│       └── requirements.txt             # Dependencies
├── paper/
│   └── Sudoma_O_Nov2025_fractional_laplacian.pdf  # Latest version
├── LICENSE
├── README.md                            # This file
└── .gitignore

Note: E53 README contains pre-registration (hypothesis, parameters). EXPERIMENTAL_REPORT shows execution results. ANALYTICAL_REPORT provides interpretation and validation.

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Mathematical Background

Fractional Laplacian generalizes classical Laplacian to non-integer orders s ∈ (0,1):

Δˢ = (-Δ)ˢ

On curved manifolds (M,g), coordinate invariance requires spectral construction:

Δˢ f = Σᵢ λᵢˢ ⟨f, φᵢ⟩ φᵢ

where {(λᵢ, φᵢ)} are eigenpairs of the Laplace-Beltrami operator.

Novel phenomena:

• Coordinate invariance holds on curved spaces
• Anomalous diffusion: ⟨r²⟩ ∝ t^α with α < 1 (subdiffusive)
• Spectral discretization naturally handles manifold geometry
• Physical applications: Diffusion in curved spacetime

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Reproducibility

All results are computationally verified:

1. Run individual experiments: See Quick Start sections above
2. Run tests: python3 -m pytest tests/ -v (in each experiment directory)
3. Expected runtime: < 10 seconds total (both experiments)

All numerical results match paper claims to stated precision.

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Citation

@misc{sudoma2025fractional,
  author = {Sudoma, Oksana},
  title = {Fractional Laplacians on Curved Manifolds: Spectral Definition, Curvature Corrections, and Anomalous Diffusion},
  year = {2025},
  doi = {10.5281/zenodo.17585575},
  url = {https://github.com/boonespacedog/Fractional-Laplacian}
}

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License

MIT License - see LICENSE file for details.

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Author

Oksana Sudoma - Independent Researcher

Computational validation and mathematical formalism assisted by Claude (Anthropic). All scientific conclusions and theoretical insights are the author's sole responsibility.

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Links

• Repository: https://github.com/boonespacedog/Fractional-Laplacian
• Zenodo Archive: https://doi.org/10.5281/zenodo.17585575
• Paper: See paper/ directory for latest version (v7)

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Physical Applications

The fractional Laplacian on curved manifolds has applications in:

• Anomalous diffusion in curved spacetime
• Non-local field theories in general relativity
• Quantum field theory on curved backgrounds
• Statistical physics on non-Euclidean geometries

This work provides the first coordinate-invariant construction with computational validation.