{"id":28766774,"url":"https://github.com/arcticoder/lqg-volume-kernel-catalog","last_synced_at":"2026-04-29T14:04:03.102Z","repository":{"id":295539392,"uuid":"990372420","full_name":"arcticoder/lqg-volume-kernel-catalog","owner":"arcticoder","description":"A comprehensive Diophantine characterization and exact kernel catalog of the four-valent Loop Quantum Gravity volume operator, including Python scripts for spectral analysis, figure generation, and automated LaTeX table output.","archived":false,"fork":false,"pushed_at":"2025-05-27T05:51:23.000Z","size":126,"stargazers_count":0,"open_issues_count":0,"forks_count":0,"subscribers_count":0,"default_branch":"main","last_synced_at":"2025-05-27T06:35:59.037Z","etag":null,"topics":["diophantine","kernel-characterization","latex","loop-quantum-gravity","lqg","python","quantum-gravity","spectral-analysis","spin-networks","sympy","volume-operator"],"latest_commit_sha":null,"homepage":"","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":"2025-05-26T02:46:55.000Z","updated_at":"2025-05-27T05:51:27.000Z","dependencies_parsed_at":"2025-05-27T06:36:00.401Z","dependency_job_id":null,"html_url":"https://github.com/arcticoder/lqg-volume-kernel-catalog","commit_stats":null,"previous_names":["arcticoder/lqg-volume-operator-spectral","arcticoder/lqg-volume-kernel-catalog"],"tags_count":0,"template":false,"template_full_name":null,"purl":"pkg:github/arcticoder/lqg-volume-kernel-catalog","repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/arcticoder%2Flqg-volume-kernel-catalog","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/arcticoder%2Flqg-volume-kernel-catalog/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/arcticoder%2Flqg-volume-kernel-catalog/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/arcticoder%2Flqg-volume-kernel-catalog/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/arcticoder","download_url":"https://codeload.github.com/arcticoder/lqg-volume-kernel-catalog/tar.gz/refs/heads/main","sbom_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/arcticoder%2Flqg-volume-kernel-catalog/sbom","host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":260355377,"owners_count":22996466,"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":["diophantine","kernel-characterization","latex","loop-quantum-gravity","lqg","python","quantum-gravity","spectral-analysis","spin-networks","sympy","volume-operator"],"created_at":"2025-06-17T12:04:23.173Z","updated_at":"2026-04-29T14:04:03.097Z","avatar_url":"https://github.com/arcticoder.png","language":"Python","funding_links":[],"categories":[],"sub_categories":[],"readme":"# LQG Volume Operator Spectral Analysis\n\nThis repository contains scripts and tools for computing and analyzing the spectrum of the Loop Quantum Gravity (LQG) volume operator and its kernel (zero-volume) states. The computations and reported counts below reflect the sampling and parameter ranges used by the included scripts; they are not exhaustive proofs of global properties.\n\n**📖 [View the complete research documentation on GitHub Pages](https://arcticoder.github.io/lqg-volume-kernel-catalog/)**\n\n## Repository Name\n\n**lqg-volume-kernel-catalog**\n\n## Description\n\nScripts to:\n- Compute the LQG volume operator spectrum via SU(2) 12j-symbols.\n- Identify trivial and non-trivial zero-volume states for 4-valent nodes.\n- Analyze kernel dimensions of the volume-squared matrix.\n- Generate LaTeX tables and figures summarizing results.\n- Validate the Diophantine characterization of zero-volume states.\n\n## Topics\n\n`loop-quantum-gravity` `volume-operator` `quantum-gravity` `su2` `recoupling-theory` `12j-symbols` `latex`\n\n## Requirements\n\nSee `requirements.txt` for dependencies:\n```bash\npip install -r requirements.txt\n```\n\n## Usage\n\n### 1. Analyze Zero-Volume States (4-valent)\n\n```bash\npython scripts/analyze_zero_volume_states.py\n```\n- Scans a specified spin-configuration range (default j=0.5 to 3.0). Adjust the script arguments to explore other ranges.\n- Identifies trivial zero-volume states under the tested sampling and criteria (J₁₂ ∩ J₃₄ = ∅ where applicable).\n- Reports that no non-trivial zero-volume states were found within the tested range and parameter sweep; this is an empirical result for the scanned configurations (not a mathematical proof of absence).\n- Outputs summary statistics and generates:\n  - `results/kernel_dimension_distribution.png`\n  - `results/zero_volume_catalog.json`\n\n### 2. Generate LaTeX Table of Trivial Zero-Volume States\n\n```bash\npython scripts/generate_latex_table.py\n```\n- Produces `results/trivial_zero_volume_table.tex` with a single-column summary of trivial zero-volume cases.\n\n### 3. Create Spin-1/2 Correlation Placeholder Figure\n\n```bash\npython scripts/generate_spin_half_correlation.py\n```\n- Generates `results/figures/spin_half_correlation.png` showing the absence of non-trivial kernel states.\n\n### 4. Generate Volume Spectrum (Optional)\n\n```bash\npython scripts/compute_volume_spectrum.py\n```\n- Computes eigenvalues of the volume-squared operator for specified spin configurations.\n- Outputs to `data/volume_spectrum.csv`.\n\n## Results\n\n- **Total configurations scanned (example run)**: 1,296 (default j_i ∈ {0.5, 1.0, …, 3.0}) — changeable via script arguments. These counts reflect the example run's sampling choices and are not exhaustive.\n- **Trivial zero-volume states (example run)**: 60 (≈4.6%) — fraction observed in the example sweep; this varies with sampling and parameterization. When reporting these counts, publish the raw outputs and sampling metadata.\n- **Non-trivial zero-volume states (example run)**: 0 observed in the example sweep (empirical result within the tested parameter range). Absence in a finite sampled set is not a mathematical proof of absence; for broader claims provide denser sampling and analytic checks.\n- **Full-rank matrices (example run)**: 674 (≈52.0%) — reported for the example sweep.\n- **Other kernel matrices (example run)**: 562 (≈43.4%) — reported for the example sweep.\n\nAll trivial zero-volume cases in the example sweep satisfy the Diophantine condition for the tested configurations:\n\n```\nmax(|j₁−j₂|, |j₃−j₄|) \u003e min(j₁+j₂, j₃+j₄)\n```\n\n## Scope, Validation \u0026 Limitations\n\n- Scope: example numerical scans of the LQG volume operator kernel for 4-valent nodes within user-configurable spin ranges. Results are empirical and depend on the sampled range and discretization; treat summary counts as illustrative of the tested parameter set.\n- Validation: run `python scripts/analyze_zero_volume_states.py --help` to see runtime options. To reproduce example results, run the script with the default parameters and attach the generated `results/` artifacts (JSON/CSV + plotting scripts) and the runtime metadata (seed, environment, args).\n- Limitations: the scripts perform finite, discrete scans. Absence of a configuration in the sampled set does not constitute a mathematical proof of absence. For stronger claims, increase sampling density, include symbolic or analytic checks where possible, and publish sensitivity analyses demonstrating robustness to sampling choices.\n\n## Directory Structure\n\n```\n├── scripts/\n│   ├── analyze_zero_volume_states.py\n│   ├── generate_latex_table.py\n│   ├── generate_spin_half_correlation.py\n│   ├── compute_volume_spectrum.py\n│   └── ...\n├── results/\n│   ├── trivial_zero_volume_table.tex\n│   ├── zero_volume_catalog.json\n│   ├── kernel_dimension_distribution.png\n│   └── figures/\n│       └── spin_half_correlation.png\n├── data/\n│   └── volume_spectrum.csv\n├── README.md\n└── requirements.txt\n```\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Farcticoder%2Flqg-volume-kernel-catalog","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Farcticoder%2Flqg-volume-kernel-catalog","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Farcticoder%2Flqg-volume-kernel-catalog/lists"}