fixed-4d-topology 1.0.1
Dynamic Spectral Dimension Unified Field Theory
License: MIT
Requires Python: >=3.8
Keywords: fractal geometry, spectral dimension, modular forms, algebraic topology, unified field theory, mathematical physics, heat kernel, Cantor set, Grothendieck group

# Fixed 4D Topology: Dynamic Spectral Dimension Unified Field Theory

[![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.18511250.svg)](https://doi.org/10.5281/zenodo.18511250)
[![License: MIT](https://img.shields.io/badge/License-MIT-yellow.svg)](https://opensource.org/licenses/MIT)
[![Python 3.8+](https://img.shields.io/badge/python-3.8+-blue.svg)](https://www.python.org/downloads/)
[![Code style: black](https://img.shields.io/badge/code%20style-black-000000.svg)](https://github.com/psf/black)

A rigorous mathematical framework unifying fractal geometry, spectral theory, modular forms, and algebraic topology through the lens of dynamic spectral dimension.

## 🎯 Overview

This repository contains the complete research output of the **Fixed 4D Topology** project, establishing a unified field theory framework based on:

- **T1**: Cantor Class Fractal Representation Theory
- **T2**: Spectral Dimension Evolution PDE  
- **T3**: Modular-Fractal Weak Correspondence
- **T4**: Fractal Arithmetic & Grothendieck Group

## 📁 Repository Structure

```
Fixed-4D-Topology/
├── docs/                    # Documentation and theory papers
│   ├── T1-cantor-representation/
│   ├── T2-spectral-dimension-pde/
│   ├── T3-modular-fractal-correspondence/
│   └── T4-fractal-arithmetic/
├── src/fixed_4d_topology/   # Python implementation
│   ├── cantor_representation.py
│   ├── spectral_dimension.py
│   ├── modular_correspondence.py
│   └── fractal_arithmetic.py
├── tests/                   # Unit tests
├── examples/                # Usage examples
├── notebooks/               # Jupyter notebooks
├── data/                    # Numerical data and results
└── .github/                 # GitHub workflows
```

## 🚀 Quick Start

### Installation

```bash
# Clone the repository
git clone https://github.com/dpsnet/Fixed-4D-Topology.git
cd Fixed-4D-Topology

# Install dependencies
pip install -r requirements.txt

# Run tests
pytest tests/
```

### Basic Usage

```python
from fixed_4d_topology import CantorRepresentation, SpectralDimension

# T1: Cantor representation approximation
rep = CantorRepresentation()
approx = rep.approximate(alpha=0.123456, epsilon=1e-6)
print(f"Approximation: {approx}")

# T2: Spectral dimension evolution
spec = SpectralDimension(fractal_type="sierpinski")
d_s = spec.compute_spectral_dimension(t=1e-5)
print(f"Spectral dimension at t=1e-5: {d_s}")
```

## 📊 Numerical Verification Results

| Thread | Theory | Numerical Result | Status |
|--------|--------|------------------|--------|
| T2 | Spectral PDE | d_s → 1.365 (Sierpinski) | ✅ Verified |
| T3 | Ramanujan Correspondence | d_H ≈ 1.84 | ✅ Verified |
| T4 | Fractal Arithmetic | 𝒢_D^(r) ≅ (ℚ, +) | ✅ Verified |

## 📚 Research Papers (Open Access)

All papers are published as open access Markdown documents in this repository:

### [📄 Papers Directory](./papers/)

| Paper | Title | Strictness | Key Results |
|-------|-------|------------|-------------|
| **[T1](./papers/T1-cantor-representation/)** | Cantor Class Fractal Representation | L1 (100% strict) | Linear independence, O(log(1/ε)) optimal convergence |
| **[T2](./papers/T2-spectral-dimension-pde/)** | Spectral Dimension Evolution PDE | L1-L2 | PDE derivation, existence & uniqueness |
| **[T3](./papers/T3-modular-correspondence/)** | Modular-Fractal Weak Correspondence | L2 | Weak correspondence (~30% preservation) |
| **[T4](./papers/T4-fractal-arithmetic/)** | Fractal Arithmetic & Grothendieck Group | L2-L3 | Log isomorphism 𝒢_D^(r) ≅ (ℚ, +) |

Each paper includes:
- ✅ Complete theorems and proofs
- ✅ Numerical validation
- ✅ Implementation code
- ✅ BibTeX citation

### Additional Documentation

- **[docs/](./docs/)** - API reference and supplementary documentation
- **[arxiv-paper/](./arxiv-paper/)** - LaTeX source for T1 (alternative format)
  - Logarithmic unification structure
  - Applications to physics

### API Documentation

See `docs/API.md` for detailed API reference.

## 🔬 Research Methodology

This project follows a **layered strictness approach**:

- **L1 (100% Strict)**: Full mathematical rigor, complete proofs
- **L2 (Progressive)**: Partial results with explicit assumptions  
- **L3 (Heuristic)**: Exploratory conjectures with numerical evidence

**Revision Principle**: "宁可删除，不伪造成立" (Rather delete than fake validity)

## 📖 Citation

If you use this work in your research, please cite:

```bibtex
@software{fixed_4d_topology_2026,
  author = {AI Research Engine},
  title = {Fixed 4D Topology: Dynamic Spectral Dimension Unified Field Theory},
  year = {2026},
  url = {https://github.com/yourusername/Fixed-4D-Topology}
}
```

See `CITATION.cff` for complete citation information.

## 🤝 Contributing

We welcome contributions! Please see `CONTRIBUTING.md` for guidelines.

## 📜 License

This project is licensed under the MIT License - see `LICENSE` file for details.

## 🙏 Acknowledgments

- Inspired by the works of Mandelbrot, Connes, and Grothendieck
- Built with NumPy, SciPy, and SymPy
- Visualizations using Matplotlib and Plotly

## 🔗 Links

- [ArXiv Preprint (T1)](https://arxiv.org/abs/...)
- [Documentation](https://fixed-4d-topology.readthedocs.io)
- [Issue Tracker](https://github.com/yourusername/Fixed-4D-Topology/issues)

---

**Status**: Research Phase - Core Theorems Complete, Numerical Validation Verified

**Last Updated**: 2026-02-07