GF(3) Tripartite Orchestration Skill

GF(3) Tripartite Orchestration

A skill for coordinating multi-agent systems using GF(3) (Galois Field of 3 elements) conservation. Implements the core pattern: local choice χ ∈ {-1, 0, +1} determines all state transitions.

Description

This skill enables agents to:

Assign trit values (-1, 0, +1) to operations and entities
Verify GF(3) conservation: Σχ ≡ 0 (mod 3)
Coordinate triadic agent compositions
Map operations to deterministic colors via Gay.jl

Usage

Trit Assignment

When orchestrating multiple agents or operations, assign trits:

| Trit | Role | Semantics | |------|------|-----------| | +1 (PLUS) | Generator | Create, advance, produce | | 0 (ERGODIC) | Transformer | Process, maintain, equilibrate | | -1 (MINUS) | Absorber | Consume, validate, verify |

Conservation Check

For any triplet of operations to compose correctly:

trit(A) + trit(B) + trit(C) ≡ 0 (mod 3)

Valid triplets:

(+1, +1, +1) → 3 ≡ 0 ✓
(+1, 0, -1) → 0 ≡ 0 ✓
(+1, -1, 0) → 0 ≡ 0 ✓
(0, 0, 0) → 0 ≡ 0 ✓
(-1, -1, +1) → -1 ≡ 0? → -1 + 3 = 2 ✗ (invalid!)

Example: ALIFE Structural Diffing

Three orthogonal vectors for change:

| Vector | Type | Trit | Description | |--------|------|------|-------------| | α | Behavioral/State | 0 (ERGODIC) | Time evolution within fixed ontology | | β | Structural/Type | +1 (PLUS) | Mutation of code, morphology, parameters | | γ | Bridge/Coherence | -1 (MINUS) | Meta-layer mapping structure to function |

Sum: α(0) + β(+1) + γ(-1) = 0 ✓

Applications

Multi-Agent Coordination

Agent A (Explorer, +1)    - generates new possibilities
Agent B (Processor, 0)    - transforms and routes
Agent C (Validator, -1)   - verifies and absorbs

Σ = +1 + 0 + (-1) = 0 ✓

World-Hopping (Counterfactual Navigation)

World-Hopping (+1)        - explore parallel worlds
Triad-Interleave (0)      - weave between possibilities
Epistemic-Arbitrage (-1)  - exploit knowledge differentials

Σ = 0 ✓

Categorical Rewriting

DisCoPy-Monoidal (+1)     - compose diagrams
Categorical-Rewriting (0) - apply DPO/SPO rules
Graph-Grafting (-1)       - attach/detach subgraphs

Σ = 0 ✓

Color Mapping

Each trit maps to a deterministic color via Gay.jl:

| Trit | Index | Hex (seed=69) | |------|-------|---------------| | +1 | 1 | #301ADC | | 0 | 2 | #7330AD | | -1 | 3 | #D192DD |

For GF(3)² (9 colors), use indices 0-8:

(0,0)=#301ADC  (0,+)=#7330AD  (0,-)=#D192DD
(+,0)=#D9DE86  (+,+)=#DD77D4  (+,-)=#E1798E
(-,0)=#A80EA2  (-,+)=#EB39A7  (-,-)=#A724AB

Scripts

Julia Validation

julia scripts/validate.jl

miniKanren Relations

guile -l scripts/gf3-kanren.scm

References

GF(3) Conservation: GADGET-SYNTHESIS.md
ALIFE Structural Diffing: ALIFE-STRUCTURAL-DIFFING.md
Gay.jl Deterministic Coloring: https://github.com/plurigrid/Gay.jl
Powers (1973) - Behavior: The Control of Perception

Author

Generated via GF(3) Tripartite Orchestration skill.

Scientific Skill Interleaving

This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:

Graph Theory

networkx [○] via bicomodule
- Universal graph hub

Bibliography References

general: 734 citations in bib.duckdb

Cat# Integration

This skill maps to Cat# = Comod(P) as a bicomodule in the equipment structure:

Trit: 0 (ERGODIC)
Home: Prof
Poly Op: ⊗
Kan Role: Adj
Color: #26D826

GF(3) Naturality

The skill participates in triads satisfying:

(-1) + (0) + (+1) ≡ 0 (mod 3)

This ensures compositional coherence in the Cat# equipment structure.

Agent Skills: GF(3) Tripartite Orchestration

Install this agent skill to your local

Skill Files