Sheaf Theoretic Coordination

# Sheaf-Theoretic Coordination

**Category:** Phase 3 Core - Distributed Reasoning
**Status:** Skeleton Implementation
**Dependencies:** `oriented-simplicial-networks`, `categorical-composition`

## Overview

Implements sheaf-theoretic coordination mechanisms for multi-agent systems, using sheaf Laplacians for consensus, harmonic extension for inference, and cohomology for detecting global obstructions.

## Capabilities

- **Sheaf Laplacian**: Consensus dynamics on cellular sheaves
- **Harmonic Extension**: Infer missing data via global consistency
- **Cohomology Detection**: Identify obstructions to global agreement
- **Sheaf Neural Networks**: Learn sheaf structures from data

## Core Components

1. **Cellular Sheaf Builder** (`cellular_sheaf.jl`)
   - Construct sheaves over cell complexes
   - Define restriction maps between stalks
   - Compute sheaf cohomology groups

2. **Sheaf Laplacian** (`sheaf_laplacian.jl`)
   - Weighted Laplacian on sheaf sections
   - Consensus dynamics and heat flow
   - Spectral analysis for convergence

3. **Harmonic Extension** (`harmonic_extension.jl`)
   - Solve for globally consistent assignments
   - Handle partial observations
   - Regularized least-squares formulation

4. **Sheaf Neural Networks** (`sheaf_nn.jl`)
   - Learn restriction maps via gradient descent
   - Sheaf diffusion layers
   - Integration with geometric deep learning

## Integration Points

- **Input from**: `oriented-simplicial-networks` (base simplicial complex)
- **Output to**: `emergent-role-assignment` (coordination constraints)
- **Coordinates with**: `categorical-composition` (sheaf functoriality)

## Usage

```julia
using SheafTheoreticCoordination

# Build cellular sheaf over graph
graph = SimplexGraph(adjacency_matrix)
sheaf = CellularSheaf(graph, stalk_dim=3)

# Define restriction maps (can be learned)
for edge in edges(graph)
    sheaf.restrictions[edge] = random_orthogonal_matrix(3)
end

# Solve for harmonic extension (inference)
partial_observations = Dict(1 => [1.0, 0.0, 0.0], 5 => [0.0, 1.0, 0.0])
global_assignment = harmonic_extension(sheaf, partial_observations)

# Check for cohomological obstructions
obstruction = compute_obstruction_cocycle(sheaf, global_assignment)
```

## References

- Hansen & Ghrist "Toward a Spectral Theory of Cellular Sheaves" (2019)
- Bodnar et al. "Sheaf Neural Networks" (ICLR 2022)
- Robinson "Topological Signal Processing" (2014)

## Implementation Status

- [x] Basic sheaf data structures
- [x] Sheaf Laplacian construction
- [ ] Full cohomology computation
- [ ] Neural sheaf learning
- [ ] Multi-agent coordination demo