Walker-Coin Lattice Theory (WCLT)

Emergent linearized Einstein gravity from discrete-time quantum walks with SU(4) directional coins.

This repository contains numerical code and data for the WCLT research program, which derives gravitational and matter structure from a minimal set of axioms: a D-dimensional lattice, a quantum walker, SU(2^{D/2}) coin operators, and locality.

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Paper 1

"Emergent Einstein Gravity from Discrete-Time Quantum Walks with SU(4) Coin Operators"
Chia-Wei Chiang — April 2026

Key Results

• Linearized Einstein gravity emerges from the Wilson plaquette action on SU(4) directional coins, with no metric or Einstein–Hilbert action imposed
• Palatini reduction of the 30×30 Hessian yields massless spin-2 dispersion (CV < 6%), Newtonian potentials Φ/Ψ ≈ 1, and 2 candidate propagating modes
• 4D one-loop fermion determinant generates a torsion mass gap: per-site splitting Δm² = +0.261 (N=4), +0.039 (N=6), +0.0325 (N=8); continuum extrapolation Δm²_cont ≈ 0.024–0.078 (lattice units)
• Floquet vacuum energy ε_vac = π/2 exactly, independent of coin angle and spatial dimension

Code

File	Description	Runtime
paper1/run_all_final.py	Core test suite (24 tests, §§2–8)	~16 min (CPU)
paper1/wclt_N8_hessian_mac.py	N=6,8 one-loop Hessian (§8.4, §8.4.7)	~5 hr (Apple Silicon)
paper1/run_scaling_study.py	Continuum scaling study (§3.6)	~5 hr (CPU)
paper1/figure_1_N_scaling.py	Reproduces Figure 1	~1 min
paper1/N8_dispersion.txt	Raw N=8 dispersion data	—

Quick Start

# Clone the repo
git clone https://github.com/chiangjw90/wclt.git
cd wclt/paper1

# Install dependencies
pip install numpy scipy matplotlib

# Run the core test suite (~16 minutes)
python run_all_final.py

# Fast mode (~3 minutes, skips 3+1D Hessian and 4D one-loop)
python run_all_final.py --fast

# Reproduce Figure 1
python figure_1_N_scaling.py

Hardware Requirements

Script	Minimum	Recommended
run_all_final.py	Any CPU, 4 GB RAM	Any modern CPU
wclt_N8_hessian_mac.py	16 GB RAM	Apple Silicon M1/M2/M3/M4 (MPS)
run_scaling_study.py	Any CPU, 4 GB RAM	Fast multi-core CPU

For GPU acceleration on Apple Silicon:

pip install torch  # PyTorch with MPS backend
python wclt_N8_hessian_mac.py --N 8 --backend torch_mps

Dependencies

numpy >= 1.20
scipy >= 1.7
matplotlib >= 3.5

Optional (for GPU acceleration):

torch >= 2.0   # PyTorch with MPS (Apple Silicon)

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Repository Structure

wclt/
└── paper1/
    ├── run_all_final.py          # 24-test suite (§§2–8)
    ├── wclt_N8_hessian_mac.py    # N=6,8 one-loop Hessian
    ├── run_scaling_study.py      # Continuum scaling (§3.6)
    ├── figure_1_N_scaling.py     # Figure 1 reproduction
    ├── N8_dispersion.txt         # Raw N=8 data
    └── requirements.txt

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Citation

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

@misc{chiang2026wclt,
  author       = {Chiang, Chia-Wei},
  title        = {Emergent Einstein Gravity from Discrete-Time Quantum Walks 
                  with SU(4) Coin Operators},
  year         = {2026},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.19653137},
  url          = {https://doi.org/10.5281/zenodo.19653137}
}

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License

• Code: MIT License
• Paper: CC-BY 4.0

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Contact

For questions or discussions, please open a GitHub Issue.