# 1-2-3-4 Pauli Model

## Overview

Maps ontological primitives to the su(2) Lie algebra (Pauli matrices).

## The Four Primitives

| # | Name | Symbol | Pauli Matrix | Interpretation |
|---|------|--------|--------------|----------------|
| 1 | Structure | Û | σ_z | What exists, identity, being |
| 2 | Change | Ĉ | σ_x | Transformation, becoming |
| 3 | Scale | L̂ | σ_y | Relation, proportion, context |
| 4 | Strength | Ŝ | iI | Emergent intensity, scalar invariant |

## Algebraic Structure

The first three form an su(2) algebra:
```
[Û, Ĉ] = 2iL̂
[Ĉ, L̂] = 2iÛ  
[L̂, Û] = 2iĈ
```

The fourth emerges from their triple product:
```
Û · Ĉ · L̂ = iI = Ŝ
```

## Interpretation

- **1-2-3** are the generators (directions of change)
- **4** is the invariant (what's preserved under change)
- Together: complete description of any dynamic system

## Connection to Physics

The fine structure constant α ≈ 1/137 may encode the relationship between these primitives and electromagnetic coupling.

## Applications to BlackRoad

1. **Agent State**: Each agent has (Û, Ĉ, L̂, Ŝ) attributes
2. **Coherence Metrics**: Measure alignment via inner products
3. **Contradiction Resolution**: Pauli algebra handles non-commuting observables

## Open Questions

- Why su(2) and not a larger algebra?
- What is the physical meaning of Ŝ?
- How does this connect to spinor geometry?