plurigrid 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.
Tags
gf3 tripartite multi-agent coordination conservation coloring
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
SDF Interleaving
This skill connects to Software Design for Flexibility (Hanson & Sussman, 2021):
Primary Chapter: 10. Adventure Game Example
Concepts: autonomous agent, game, synthesis
GF(3) Balanced Triad
gf3-tripartite (○) + SDF.Ch10 (+) + [balancer] (−) = 0
Skill Trit: 0 (ERGODIC - coordination)
Secondary Chapters
- Ch1: Flexibility through Abstraction
- Ch4: Pattern Matching
- Ch6: Layering
Connection Pattern
Adventure games synthesize techniques. This skill integrates multiple patterns.
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.
Para(Optic) atlas
Part of: para-mensch-commons.