Tetrahedron Implementation Plan

Created: 2025-05-24

Two-Phase Approach

Phase 1: Basic Tetrahedron (T67 - Ready Now)

What: Sequential coupling of 4 spins using existing T66 capability Result: Valid quantum state, but fixed coupling order ((j1⊗j2)⊗j3)⊗j4 Limitation: Cannot transform between coupling schemes

interface BasicTetrahedron {
  edgeSpins: number[];        // [j1, j2, j3, j4] 
  coupledState: StateVector;  // Final result
  totalJ: number;             // Effective angular momentum
}

Phase 2: Complete Tetrahedron (Requires 6j Symbols)

What: Full intertwiner space with recoupling transformations Result: All possible tetrahedron states, proper geometric symmetries Requirement: 6j symbols for transforming between coupling schemes

interface CompleteTetrahedron extends BasicTetrahedron {
  intertwiners: number[];           // All allowed J values
  basisStates: StateVector[];       // Complete orthogonal basis
  recouplingTransforms: Matrix[];   // 6j-based transformations
}

Why 6j Symbols are Essential

Research shows 6j symbols are "the basic building block" for quantum tetrahedra because:

1. Handle coupling of 3+ angular momenta with scheme transformations
2. Provide unitary transforms between different coupling orders
3. Enable proper geometric interpretation of quantum tetrahedra

Implementation Timeline

• Phase 1: 3-4 days (can start immediately with T66)
• 6j Symbols: 3-4 days (extends T55a Wigner module)
• Phase 2: 4-5 days (after 6j symbols complete)

Bottom Line

T67 can start now with basic construction. Complete tetrahedron needs 6j symbols for mathematical rigor and physical accuracy.