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sheaf-theoretic-coordination skill

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This skill enables distributed coordination using sheaf theory to achieve consensus, infer missing data, and detect obstructions in multi-agent systems.

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SKILL.md

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---
name: "sheaf-theoretic-coordination"
description: "**Category:** Phase 3 Core - Distributed Reasoning"
---

# 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

Overview

This skill implements sheaf-theoretic coordination primitives for distributed multi-agent systems. It provides tools for building cellular sheaves on complexes, running consensus via sheaf Laplacians, performing harmonic extension to infer missing data, and detecting global obstructions via cohomology. The implementation is a skeleton focused on core data structures and Laplacian-based dynamics with planned extensions for full cohomology and learning.

How this skill works

You construct a cellular sheaf over a graph or simplicial complex by assigning vector-space stalks and linear restriction maps on faces. The sheaf Laplacian encodes local consistency constraints and drives consensus or diffusion on sections. Harmonic extension solves a regularized least-squares problem to infer globally consistent assignments from partial observations, while cohomology routines detect obstructions that prevent full agreement.

When to use it

Coordinating states across a network of heterogeneous agents with local compatibility constraints
Filling in missing sensor or agent data while enforcing geometric or relational consistency
Diagnosing global inconsistencies or incompatibilities that block consensus
Embedding topological constraints into learning pipelines for distributed systems
Analyzing convergence and spectral properties of constrained consensus dynamics

Best practices

Model agent variables as vector-space stalks and express constraints as linear restriction maps
Normalize weights and restriction maps to improve Laplacian conditioning and spectral behavior
Provide reliable partial observations on a well-distributed set of nodes to enable stable harmonic extension
Use spectral analysis of the sheaf Laplacian to select regularization and step sizes for iterative solvers
Validate cohomology checks on small subsystems before scaling to the full network

Example use cases

Distributed sensor networks: infer missing sensor readings while respecting calibration relations between neighbors
Multi-robot coordination: enforce compatible local frames or control parameters using sheaf constraints
Consensus with heterogeneous state dimensions: combine agents with different state sizes using tailored restriction maps
Fault detection: detect and localize global obstructions that indicate inconsistent configurations
Hybrid pipelines: learn restriction maps from data using sheaf neural layers and then run harmonic extension for inference

FAQ

What does the sheaf Laplacian provide compared to a graph Laplacian?

The sheaf Laplacian incorporates local linear constraints between vector-valued stalks, enabling consensus over heterogeneous state spaces rather than scalar signals.

Can harmonic extension recover arbitrary missing data?

Harmonic extension recovers data consistent with the sheaf constraints; recovery quality depends on where observations are placed and whether cohomological obstructions exist.