Unified Closed-Form Representations and Generating Functionals for SU(2) 3n-j Recoupling Coefficients
Bibliographic record
Abstract
We present a unified framework for SU(2) 3n-j recoupling coefficients (Wigner symbols) encompassing five complementary closed-form representations. Our central result is a hypergeometric product formula valid for arbitrary trivalent coupling graphs, expressed via graph matching numbers and ${}_2F_1$ functions. This unifies all special cases (6j, 9j, 12j, etc.) under a single computational framework. The framework comprises: (1) hypergeometric product formulas achieving polynomial complexity $O(|E| \cdot j^2)$ versus exponential summation methods; (2) uniform single-sum ${}_5F_4$ representations for 12j symbols via algebraic reindexing; (3) finite three-term recurrence relations from edge-removal determinants with documented stability regimes; (4) universal Schwinger-boson generating functionals yielding determinant formulas $G(\{x_e\}) = 1/\sqrt{\det(I-K)}$; and (5) arbitrary-valence node matrix elements via functional derivatives. All representations are rigorously cross-validated through 178 automated tests (21 integration checks across five independent implementations, plus 157 per-repository unit tests), with exact agreement against SymPy's symbolic computation. High-precision (50 decimal places) reference datasets for 6j, 9j, and 12j symbols establish deterministic validation baselines. We provide comprehensive uncertainty quantification protocols documenting numerical stability regimes, precision requirements, and failure modes for practical computation. Complete source code, validation scripts, and reference datasets are available in open-source repositories.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.010 | 0.003 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".