{"id":28945310,"url":"https://github.com/arcticoder/su2-3nj-uniform-closed-form","last_synced_at":"2025-10-27T03:11:59.233Z","repository":{"id":295500826,"uuid":"990272513","full_name":"arcticoder/su2-3nj-uniform-closed-form","owner":"arcticoder","description":"Universal closed-form hypergeometric representation of SU(2) 3nj symbols, featuring a LaTeX derivation, master generating functional, and Python scripts for symbolic expansion and numerical verification.","archived":false,"fork":false,"pushed_at":"2025-06-21T14:13:56.000Z","size":174,"stargazers_count":1,"open_issues_count":0,"forks_count":0,"subscribers_count":0,"default_branch":"main","last_synced_at":"2025-06-21T14:38:32.000Z","etag":null,"topics":["3nj","computational-physics","generating-function","hypergeometric","latex","mathematics","mathjax","orthogonal-polynomials","physics","quantum-mechanics","recoupling","su2","symbolic-computation","sympy"],"latest_commit_sha":null,"homepage":"https://arcticoder.github.io/su2-3nj-uniform-closed-form/","language":"Python","has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":null,"status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/arcticoder.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":null,"code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null,"zenodo":null}},"created_at":"2025-05-25T21:01:38.000Z","updated_at":"2025-06-21T14:14:00.000Z","dependencies_parsed_at":"2025-05-25T22:40:28.634Z","dependency_job_id":null,"html_url":"https://github.com/arcticoder/su2-3nj-uniform-closed-form","commit_stats":null,"previous_names":["arcticoder/su2-3nj-uniform-closed-form"],"tags_count":0,"template":false,"template_full_name":null,"purl":"pkg:github/arcticoder/su2-3nj-uniform-closed-form","repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/arcticoder%2Fsu2-3nj-uniform-closed-form","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/arcticoder%2Fsu2-3nj-uniform-closed-form/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/arcticoder%2Fsu2-3nj-uniform-closed-form/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/arcticoder%2Fsu2-3nj-uniform-closed-form/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/arcticoder","download_url":"https://codeload.github.com/arcticoder/su2-3nj-uniform-closed-form/tar.gz/refs/heads/main","sbom_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/arcticoder%2Fsu2-3nj-uniform-closed-form/sbom","host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":261434233,"owners_count":23157210,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2022-07-04T15:15:14.044Z","host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":["3nj","computational-physics","generating-function","hypergeometric","latex","mathematics","mathjax","orthogonal-polynomials","physics","quantum-mechanics","recoupling","su2","symbolic-computation","sympy"],"created_at":"2025-06-23T07:10:25.368Z","updated_at":"2025-10-27T03:11:59.227Z","avatar_url":"https://github.com/arcticoder.png","language":"Python","funding_links":[],"categories":[],"sub_categories":[],"readme":"# Universal Closed-Form Hypergeometric Representation of SU(2) 3nj Symbols\n\n[![GitHub Pages](https://img.shields.io/badge/GitHub%20Pages-Live-brightgreen)](https://arcticoder.github.io/su2-3nj-uniform-closed-form/)\n[![License](https://img.shields.io/badge/License-MIT-blue.svg)](LICENSE)\n[![Python](https://img.shields.io/badge/Python-3.7+-blue.svg)](https://python.org)\n\nThis 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.\n\n## Mathematical Framework\n\nThe 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:\n\n- Symbolic derivations are provided for a range of topologies; some edge cases may require additional analysis.\n- The framework can offer alternative evaluation approaches for certain symbol classes; users should benchmark against existing libraries for their specific parameter ranges.\n- Derivations and proofs are included in the LaTeX source; consult the paper for assumptions and scope.\n- Numerical verification is provided for selected cases but is not a comprehensive validation across all spins and couplings.\n\n## 📖 Contents\n\n- **LaTeX Source**: Mathematical derivations and supporting notes\n- **GitHub Pages Website**: Interactive exposition with MathJax rendering\n- **PDF Documentation**: Publication-ready mathematical exposition\n- **Computational Scripts**: Python implementation and verification tools\n- **Validation Data**: Numerical verification results and benchmarks for tested cases\n\n## 🌐 Online Documentation\n\n**📚 Read the paper online**: https://arcticoder.github.io/su2-3nj-uniform-closed-form/\n\nThe website features interactive exposition, downloadable PDF, example code, and links to related work in the SU(2) 3nj series.\n\n## Computational Verification\n\nThe repository includes symbolic and numeric scripts intended as reproducibility artifacts for the included examples.\n\n### 🔄 Symbolic Taylor Expansion\n**Script**: `symbolic_taylor_expansion.py`\n- Generates symbolic series for illustrative cases and inspects coefficients for internal consistency.\n\n### Hypergeometric Correspondence\n**Script**: `match_simplest_hypergeometric.py`\n- Demonstrates correspondence with known 9j symbol representations for selected parameter choices.\n\n### Numerical Validation\n**Primary**: `verify_simple_9j_numeric.py`\n- High-precision numeric checks for representative simple cases.\n\n**Extended**: `verify_additional_9j_numeric.py`\n- Additional numeric checks across a small set of cases; intended as a starting point for broader validation.\n\n**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.\n\n## Installation \u0026 Usage\n\n### Prerequisites\n```bash\npip install sympy numpy scipy pandas matplotlib\n```\n\n### Running Verification Scripts\n```bash\n# Symbolic Taylor expansion\npython symbolic_taylor_expansion.py\n\n# Hypergeometric matching\npython match_simplest_hypergeometric.py\n\n# Numerical validation\npython verify_simple_9j_numeric.py\npython verify_additional_9j_numeric.py\n```\n\n## 🔗 Related Work\n\nThis repository is part of an SU(2) 3nj symbol research series:\n\n- `su2-3nj-closedform`: Closed-form hypergeometric product formula\n- `su2-3nj-recurrences`: Finite closed-form recurrence relations\n- `su2-3nj-generating-functional`: Generating functional approach\n- `su2-node-matrix-elements`: Operator matrix elements for arbitrary-valence nodes\n\n## Mathematical Background\n\n### Core Theory\nThe 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.\n\n### Key Considerations\n- The generating functional is proposed as a compact formal expression covering many topologies under stated assumptions.\n- Derived closed-form expressions are provided for several topologies; additional cases may require extended derivations or boundary data.\n- Performance and numerical stability depend on parameter ranges and chosen numerical precision; validate against established implementations for production use.\n\n## Applications\n\n- Quantum mechanics: angular momentum coupling computations (research/analysis use)\n- Computational physics: experimental evaluation of evaluation techniques\n- Mathematical physics: special-function identities and illustrative examples\n\n## License\n\nThis project is licensed under The Unlicense - see the `LICENSE` file for details.\n\n## Contributing\n\nContributions are welcome; for major changes please open an issue first to discuss the proposal.\n\n---\n\n## Scope / Validation \u0026 Limitations\n\n- **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.\n- **Numerical stability \u0026 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.\n- **Uncertainty \u0026 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.\n- **Limitations:** Derivations assume the conditions stated in the paper; edge cases (boundary conditions, degenerate couplings) may require additional analysis or boundary data.","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Farcticoder%2Fsu2-3nj-uniform-closed-form","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Farcticoder%2Fsu2-3nj-uniform-closed-form","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Farcticoder%2Fsu2-3nj-uniform-closed-form/lists"}