gauge-nn-interpretability 1.0.3
Computational Gauge Theory for Neural Network Interpretability
Requires Python: >=3.8
Keywords: neural-networks interpretability gauge-theory transformers machine-learning

# Computational Gauge Theory for Neural Network Interpretability

## Breaking the Black Box: A Mathematical Framework for Transparent AI

This framework treats transformers and neural networks not as opaque black boxes, but as **computational gauge fields** with rigorous mathematical guarantees for reasoning, performance, and interpretability.

## 🎯 Core Innovation

We shift from heuristics to **first principles** by treating neural network operations as elements of non-abelian Lie algebras, where:

- **Curvature** measures non-commutativity of reasoning paths
- **Commutators** [A,B] = AB - BA reveal how order matters in attention
- **BCH plaquettes** expose higher-order algebraic structure
- **Jacobiators** J(A,B,C) capture hierarchical thinking
- **Wilson loops** trace reasoning chains with path-dependent evolution
- **Homotopy invariants** guarantee stable generalization

## 📐 Mathematical Foundation

### Gauge Field Structure
```
F_ij = [∇_i, ∇_j] = ∂_i A_j - ∂_j A_i + [A_i, A_j]
```
where A_i are attention operators at layer i.

### Baker-Campbell-Hausdorff Formula
```
e^A e^B = e^{A + B + 1/2[A,B] + 1/12[A,[A,B]] + ...}
```
reveals compositional structure of sequential operations.

### Jacobiator (measures associativity failure)
```
J(A,B,C) = [[A,B],C] + [[B,C],A] + [[C,A],B]
```

### Wilson Loop (path-ordered product)
```
W(C) = P exp(∮_C A_μ dx^μ)
```

## 🔬 What This Framework Reveals

1. **Reasoning Chains**: Not vague activations, but loops and curvature
2. **Deep Thinking**: Measured by homotopy invariance of operator invariants
3. **Attention Structure**: Shows heads organizing into algebraic structures
4. **Abelian Collapse**: Detects when thinking becomes linear (bad)
5. **Stability Guarantees**: Symmetries ensure generalization

## 🚀 Features

- ✅ Full mathematical implementation of gauge theory operators
- ✅ Commutator, curvature, and Jacobiator calculations
- ✅ BCH expansion to arbitrary order
- ✅ Wilson loop computation for reasoning path analysis
- ✅ Homotopy invariant extraction
- ✅ Transformer-specific interpretability layer
- ✅ Real-time visualization of reasoning geometry
- ✅ Mathematical proof verification
- ✅ Works with any transformer architecture

## 📊 Applications

- **Model Debugging**: See exactly where reasoning fails
- **Architecture Design**: Optimize for non-abelian structure
- **Training Guidance**: Maximize curvature in reasoning paths
- **Explainability**: Show users the geometric path of inference
- **Safety**: Detect collapse to deterministic patterns

## 🎓 Built By

**Michael J. Pendleton**
Doctoral Candidate in AI/ML Engineering, George Washington University
CEO, The AI Cowboys
michael.pendleton.20@gmail.com

---

## Installation

```bash
pip install -r requirements.txt
```

## Quick Start

```python
from gauge_nn_interpretability import GaugeAnalyzer

# Load your transformer
analyzer = GaugeAnalyzer(model)

# Extract gauge-theoretic properties
curvature = analyzer.compute_curvature()
plaquettes = analyzer.compute_bch_plaquettes()
reasoning_loops = analyzer.trace_wilson_loops()

# Visualize
analyzer.visualize_reasoning_geometry()
```

## Citation

If you use this framework, please cite:

```bibtex
@software{pendleton2025gauge,
  author = {Pendleton, Michael J.},
  title = {Computational Gauge Theory for Neural Network Interpretability},
  year = {2025},
  url = {https://github.com/theaicowboys/gauge-nn-interpretability}
}
```

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**Bridge from heuristics to physics. From "does it work?" to WHY it works.**