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consensus skill

/skills/consensus

This skill analyzes and enforces consensus protocols in multi-agent systems, guiding stable agreement and robust dynamical behavior across agents.

npx playbooks add skill plurigrid/asi --skill consensus

Review the files below or copy the command above to add this skill to your agents.

Files (1)
SKILL.md
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---
name: consensus
description: Agreement protocol in multi-agent systems
trit: 0
geodesic: true
moebius: "μ(n) ≠ 0"
---

# Consensus

**Trit**: 0 (ERGODIC)
**Domain**: Dynamical Systems Theory
**Principle**: Agreement protocol in multi-agent systems

## Overview

Consensus is a fundamental concept in dynamical systems theory, providing tools for understanding the qualitative behavior of differential equations and flows on manifolds.

## Mathematical Definition

```
CONSENSUS: Phase space × Time → Phase space
```

## Key Properties

1. **Local behavior**: Analysis near equilibria and invariant sets
2. **Global structure**: Long-term dynamics and limit sets  
3. **Bifurcations**: Parameter-dependent qualitative changes
4. **Stability**: Robustness under perturbation

## Integration with GF(3)

This skill participates in triadic composition:
- **Trit 0** (ERGODIC): Neutral/ergodic
- **Conservation**: Σ trits ≡ 0 (mod 3) across skill triplets

## AlgebraicDynamics.jl Connection

```julia
using AlgebraicDynamics

# Consensus as compositional dynamical system
# Implements oapply for resource-sharing machines
```

## Related Skills

- equilibrium (trit 0)
- stability (trit +1)  
- bifurcation (trit +1)
- attractor (trit +1)
- lyapunov-function (trit -1)

---

**Skill Name**: consensus
**Type**: Dynamical Systems / Consensus
**Trit**: 0 (ERGODIC)
**GF(3)**: Conserved in triplet composition

## Non-Backtracking Geodesic Qualification

**Condition**: μ(n) ≠ 0 (Möbius squarefree)

This skill is qualified for non-backtracking geodesic traversal:

1. **Prime Path**: No state revisited in skill invocation chain
2. **Möbius Filter**: Composite paths (backtracking) cancel via μ-inversion
3. **GF(3) Conservation**: Trit sum ≡ 0 (mod 3) across skill triplets
4. **Spectral Gap**: Ramanujan bound λ₂ ≤ 2√(k-1) for k-regular expansion

```
Geodesic Invariant:
  ∀ path P: backtrack(P) = ∅ ⟹ μ(|P|) ≠ 0
  
Möbius Inversion:
  f(n) = Σ_{d|n} g(d) ⟹ g(n) = Σ_{d|n} μ(n/d) f(d)
```

Overview

This skill captures consensus protocols in multi-agent dynamical systems, focusing on how agents reach agreement over time. It frames consensus as a map from phase space and time to evolved phase states and emphasizes qualitative long-term behavior, stability, and ergodic properties. The skill highlights topological and algebraic structures that govern collective dynamics and non-backtracking traversal qualifications.

How this skill works

The skill inspects agent state trajectories, equilibrium sets, and invariant manifolds to determine local and global agreement behavior. It analyzes parameter-dependent bifurcations and stability under perturbations to predict limit sets and convergence. It also encodes triadic GF(3) composition rules and Möbius-filter conditions used to qualify non-backtracking geodesic traversals in skill chains.

When to use it

  • Designing or verifying distributed consensus algorithms for sensor networks, robotics, or multi-agent control.
  • Studying stability, attractors, and bifurcation scenarios in continuous-time or discrete-time multi-agent models.
  • Composing skills in pipelines where non-backtracking state transitions and modular conservation (GF(3)) are required.
  • Analyzing ergodic behavior or long-term qualitative outcomes of interacting dynamical subsystems.
  • Filtering invocation chains to ensure prime-path traversal using Möbius inversion constraints.

Best practices

  • Model agents on an appropriate phase space and explicitly identify invariant sets and equilibria before applying consensus analysis.
  • Check spectral gap and expansion bounds when using graph-based interaction models to guarantee convergence rates.
  • Use bifurcation checks to detect parameter regimes that change agreement outcomes and design controllers accordingly.
  • Apply Möbius-filter conditions to eliminate backtracking in composition chains and preserve GF(3) conservation in triplets.
  • Validate predictions with simulation of representative initial conditions and perturbations to assess robustness.

Example use cases

  • Verify convergence of a flocking controller where agents must reach velocity agreement despite time-varying links.
  • Analyze how changing coupling strength induces bifurcations that break consensus and create multiple limit sets.
  • Compose three modular skills whose trits must sum to zero modulo 3 to maintain conserved behavior in a larger pipeline.
  • Filter a sequence of skill invocations to ensure no state is revisited (prime path) using Möbius inversion logic.
  • Estimate convergence rates using spectral gap bounds for k-regular interaction graphs in networked control.

FAQ

What does trit 0 (ERGODIC) imply for consensus behavior?

Trit 0 marks neutral or ergodic behavior: the skill focuses on long-term qualitative dynamics without biasing toward contraction or divergence.

How does the GF(3) conservation affect composing skills?

When composing skills in triplets, ensure the sum of trits is 0 mod 3; this rule preserves the intended algebraic conservation across the composite pipeline.