Universal Closed-Form Hypergeometric Representation of SU(2) 3nj Symbols

This repository presents a mathematical framework and illustrative computations for closed-form hypergeometric representations of SU(2) 3nj symbols, with particular focus on 12j symbols and a proposed generating functional. The materials emphasize derivations and representative computational checks rather than exhaustive numerical validation across all parameter regimes.

Mathematical Framework

The project proposes a closed-form hypergeometric representation that aims to cover many SU(2) 3nj recoupling coefficients within a special-function framework. The repository documents derivations and provides representative computational checks. Key points:

• Symbolic derivations are provided for a range of topologies; some edge cases may require additional analysis.
• The framework can offer alternative evaluation approaches for certain symbol classes; users should benchmark against existing libraries for their specific parameter ranges.
• Derivations and proofs are included in the LaTeX source; consult the paper for assumptions and scope.
• Numerical verification is provided for selected cases but is not a comprehensive validation across all spins and couplings.

📖 Contents

• LaTeX Source: Mathematical derivations and supporting notes
• GitHub Pages Website: Interactive exposition with MathJax rendering
• PDF Documentation: Publication-ready mathematical exposition
• Computational Scripts: Python implementation and verification tools
• Validation Data: Numerical verification results and benchmarks for tested cases

🌐 Online Documentation

📚 Read the paper online: https://arcticoder.github.io/su2-3nj-uniform-closed-form/

The website features interactive exposition, downloadable PDF, example code, and links to related work in the SU(2) 3nj series.

Computational Verification

The repository includes symbolic and numeric scripts intended as reproducibility artifacts for the included examples.

🔄 Symbolic Taylor Expansion

Script: symbolic_taylor_expansion.py

• Generates symbolic series for illustrative cases and inspects coefficients for internal consistency.

Hypergeometric Correspondence

Script: match_simplest_hypergeometric.py

• Demonstrates correspondence with known 9j symbol representations for selected parameter choices.

Numerical Validation

Primary: verify_simple_9j_numeric.py

• High-precision numeric checks for representative simple cases.

Extended: verify_additional_9j_numeric.py

• Additional numeric checks across a small set of cases; intended as a starting point for broader validation.

Output: Results and verification artifacts are stored in data/ for the tested examples; these serve as reproducibility artifacts rather than proof of exhaustive correctness across all regimes.

Installation & Usage

Prerequisites

pip install sympy numpy scipy pandas matplotlib

Running Verification Scripts

# Symbolic Taylor expansion
python symbolic_taylor_expansion.py

# Hypergeometric matching
python match_simplest_hypergeometric.py

# Numerical validation
python verify_simple_9j_numeric.py
python verify_additional_9j_numeric.py

🔗 Related Work

This repository is part of an SU(2) 3nj symbol research series:

• su2-3nj-closedform: Closed-form hypergeometric product formula
• su2-3nj-recurrences: Finite closed-form recurrence relations
• su2-3nj-generating-functional: Generating functional approach
• su2-node-matrix-elements: Operator matrix elements for arbitrary-valence nodes

Mathematical Background

Core Theory

The universal representation is presented as a hypergeometric-based construction that relates angular-momentum coupling topologies to special-function expressions. See the paper for precise definitions and applicable assumptions.

Key Considerations

• The generating functional is proposed as a compact formal expression covering many topologies under stated assumptions.
• Derived closed-form expressions are provided for several topologies; additional cases may require extended derivations or boundary data.
• Performance and numerical stability depend on parameter ranges and chosen numerical precision; validate against established implementations for production use.

Applications

• Quantum mechanics: angular momentum coupling computations (research/analysis use)
• Computational physics: experimental evaluation of evaluation techniques
• Mathematical physics: special-function identities and illustrative examples

License

This project is licensed under The Unlicense - see the LICENSE file for details.

Contributing

Contributions are welcome; for major changes please open an issue first to discuss the proposal.

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Scope / Validation & Limitations

• Research-stage framework: Materials document a theoretical framework and illustrative verifications. Claims about universality are intended as working hypotheses derived in the paper; maintainers and users should verify applicability for new topologies and large-spin regimes.
• Numerical stability & validation: Numerical checks included here cover selected examples; reproduce these checks in your environment and extend them for other parameter ranges. Consider using high-precision arithmetic for large spins and cross-validate against established libraries.
• Uncertainty & reproducibility: When publishing numeric comparisons, include environment details, numerical precision, random seeds (if any), and input parameter sets under docs/ or data/ to support reproducibility.
• Limitations: Derivations assume the conditions stated in the paper; edge cases (boundary conditions, degenerate couplings) may require additional analysis or boundary data.