SU(2) 3n-j Unified Framework

Unified representations for SU(2) 3n-j recoupling coefficients: hypergeometric products, uniform sums, finite recurrences, generating functionals, and node-matrix elements.

Core novelty: First closed-form product over matching ratios for arbitrary trivalent graphs (Theorem 1); determinant-based functional yielding convention- independent coefficients $C_G$ (Theorem 4).

Paper: Unified Closed-Form Representations and Generating Functionals for SU(2) 3n-j Recoupling Coefficients — targeting SIGMA (primary), JPA (backup).

Validation: 161+ pytest tests; Wolfram verification: wolframscript -file scripts/verify_wolfram.wls --paper-strict (all pass).

Subdirectories

All implementations are now included directly in this repository:

Directory	Description	Tests
generating-functional/	Det(I−K) implementation — Theorem 4	43 ✓
uniform-closed-form/	Single-sum ₅F₄ for 12j — Theorem 2	45 ✓
node-matrix-elements/	Operator matrix elements — Theorem 5	24 ✓
closedform/	Hypergeometric product — Theorem 1	27 ✓
recurrences/	Three-term recurrences — Theorem 3	18 ✓
lean/	Lean 4 formal support for Theorems 1 & 4	—
papers/paper/	LaTeX source for the unified paper	—
scripts/	Validation scripts (Wolfram, Python, MATLAB)	—

Quick start

# Cross-repo Python validation
python scripts/run_integration_tests.py

# Wolfram verification
wolframscript -file scripts/verify_wolfram.wls --paper-strict

# Build paper PDF
cd papers/paper && make

Lean 4 formal proofs

The lean/ directory contains Lean 4 formalization support for the key theorems, using Mathlib 4.27.0. Theorem 1 is stated formally; supporting lemmas are proved.

cd lean && lake build

Status (2026-03-01)

Component	Status
All 5 implementations	✅ 161+ tests pass
Paper (SIGMA format)	✅ Compiles cleanly
Lean formal support	✅ Builds (lake build)
arXiv upload	⬜ Pending
Journal submission (SIGMA)	⬜ Pending