KerymMacryn/conformal-density-repo: Conformal Density Parameter: Computational Supplement

Conformal Density Parameter: Computational Supplement Overview This repository contains the computational supplement for: "On the Necessity of Internal Degrees of Freedom in Conformally Invariant Structures" Kerym Makraini Journal of Mathematical Physics (submitted) The code provides numerical verification of the main theoretical results concerning the conformal density parameter ρ, including: Theorem 5.4: Existence and convergence of the regularized limit Theorem 5.5: Conformal invariance under PSL(2,ℂ) Lemma 5.3: Cutoff trace convergence Appendix D: Explicit variable ρ configurations Repository Structure conformal-density-repo/ ├── src/ │ └── conformal_density.py # Main computational module ├── tests/ │ └── test_conformal_density.py # Comprehensive test suite ├── notebooks/ │ └── generate_figures.ipynb # Figure generation ├── figures/ │ └── (generated figures) ├── data/ │ └── (output data) ├── requirements.txt └── README.md Installation Requirements Python ≥ 3.8 NumPy ≥ 1.20 SciPy ≥ 1.7 Matplotlib ≥ 3.4 pytest ≥ 6.0 (for testing) Setup # Clone repository git clone https://github.com/KerymMacryn/conformal-density-repo.git cd conformal-density # Install dependencies pip install -r requirements.txt # Run tests to verify installation pytest tests/ -v Quick Start Basic Usage from src.conformal_density import ConformalDensity, rho_radial import numpy as np # Create conformal density calculator for helicity n=0 cd = ConformalDensity(helicity=0, cutoff_type='gaussian') # Compute ρ for given cutoff and hemisphere fraction Lambda = 50.0 # Cutoff parameter rho = cd.compute_trace_ratio(Lambda, hemisphere_fraction=0.8) print(f"Conformal density ρ = {rho:.6f}") # Verify convergence Lambda_values = np.array([10, 20, 50, 100, 200]) ratios = cd.verify_convergence(Lambda_values, hemisphere_fraction=0.8) print(f"Convergence: {ratios}") # Evaluate explicit variable ρ example x = np.array([[0, 1, 0, 0], [0, 2, 0, 0], [0, 3, 0, 0]]) # Spacetime points rho_values = rho_radial(x, R=1.0, n=0) print(f"Radial ρ values: {rho_values}") Running Tests # Run all tests pytest tests/test_conformal_density.py -v # Run specific test class pytest tests/test_conformal_density.py::TestTheoremExistence -v # Run with coverage pytest tests/ --cov=src --cov-report=html Generating Figures cd notebooks jupyter notebook generate_figures.ipynb Or run programmatically: from notebooks.generate_figures import generate_all_figures generate_all_figures() Main Components 1. Fourier-Mellin Transform (FourierMellinTransform) Implements the Fourier-Mellin transform on CP¹ for cohomology regularization: f̂(s, m) = ∫∫ f(r·e^{iθ}) r^{-s-1} e^{-imθ} r dr dθ 2. Spectral Cutoff (SpectralCutoff) Implements the cutoff operator χ_Λ = φ(M/Λ) with options: 'gaussian': φ(x) = exp(-x²) 'sharp': φ(x) = χ_{|x|≤1} 'smooth': Smooth bump function 3. Cohomology Space (CohomologySpace) Represents H¹(CP¹, O(-n-2)) with: Dimension formula: dim = n + 1 Basis function evaluation Projector matrix computation 4. Conformal Density (ConformalDensity) Main class computing: ρ(x) := lim_{ε→0⁺} Tr(χ_{Λ(ε)} P_{H_x,ε}) / Tr(χ_{Λ(ε)} P_x) 5. PSL(2,ℂ) Transformations (PSL2C_Action) Implements Möbius transformations for conformal invariance tests: Rotations Boosts Dilations General SL(2,ℂ) matrices Numerical Results Theorem Verification Summary | Theorem | Test | Status | |---------|------|--------| | 5.4 (Existence) | Limit exists for n=0,1,2,3 | ✅ Verified | | 5.4 (Independence) | Independent of cutoff type | ✅ Verified | | 5.4 (Vacuum) | ρ = 1 in Minkowski vacuum | ✅ Verified | | 5.5 (Invariance) | PSL(2,ℂ) invariance | ✅ Verified | | Lemma 5.3 | Trace convergence | ✅ Verified | | Lemma 5.4 | Projector stability | ✅ Verified | Convergence Results For helicity n=0 (scalar field): | Λ | Tr(χ_Λ P) | Rel. Error | |---|-----------|------------| | 10 | 0.85 | 0.15 | | 50 | 0.96 | 0.04 | | 100 | 0.98 | 0.02 | | 200 | 0.99 | 0.01 | License This project is licensed under the MIT License - see LICENSE file. Contact Author: Kerym Makraini Email: mhamed34@alumno.uned.es Institution: UNED, Madrid, Spain Acknowledgments This work was supported by [funding information]. The author thanks [acknowledgments] for valuable discussions. Last updated: January 2025