# Spiral Information Geometry

> Status: 🔴 Planned

## Vision

Use **information geometry** to formalize the coherence formula and agent dynamics.

## Core Concepts

| Concept | Meaning |
|---------|---------|
| Fisher Information | Metric on probability distributions |
| Natural Gradient | True direction of steepest descent |
| Geodesic | Shortest path in information space |
| Curvature | How much space "bends" around a point |

## Research Agenda

### 1. Coherence as Distance
- C(t) = geodesic distance between agent states?
- Coherent agents = nearby in information space

### 2. Contradiction as Curvature
- High δ_t = high curvature regions
- Creative energy peaks at curvature maxima

### 3. Learning as Parallel Transport
- Agent learning = transport along geodesics
- Memory = holonomy (what changes after round trip)

### 4. Partition Function as Potential
- Z = Σ e^{-βH} defines a potential landscape
- Equilibrium = potential minimum

## Connections

- **n=π Duality**: Fisher metric on discrete vs continuous?
- **Creative Energy**: K(t) related to scalar curvature?
- **Remainder Principle**: Curvature = remainder of flatness

## Next Steps

1. Formalize C(t) using Fisher-Rao metric
2. Compute geodesics for simple agent models
3. Relate curvature to δ_t experimentally

## References

- Amari, S. "Information Geometry and Its Applications"
- Nielsen, F. "An Elementary Introduction to Information Geometry"