Skill Validation GF(3) - SLAVE (-1)

"The validator constrains and verifies."

XIP Assignment

Purpose

This skill validates that all skills in the repository:

1. Follow GF(3) conservation across triads
2. Have deterministic Gay.jl colors assigned
3. Maintain role consistency (GAY/MASTER/SLAVE)

Validation Rules

Rule 1: Skill Structure

Every skill must have:

skills/<skill-name>/
├── SKILL.md           # Required
├── *.py|*.rb|*.jl     # Implementation (optional)
└── tests/             # Validation tests (optional)

Rule 2: GF(3) Triad Declaration

Skills should declare their triad membership:

## GF(3) Triad

| Role | Skill | Trit |
|------|-------|------|
| GAY (+1) | skill-a | +1 |
| MASTER (0) | skill-b | 0 |
| SLAVE (-1) | skill-c | -1 |

Sum: (+1) + (0) + (-1) = 0 ✓

Rule 3: Color Assignment

Colors must be deterministic via Gay.jl:

from gay_mcp import color_at

# Verify skill color
assert color_at(seed=2025, index=8)['hex'] == '#4857D5'

Validation Script

#!/usr/bin/env python3
"""Validate all skills for GF(3) conservation."""

import os
import re
from pathlib import Path

def validate_skill(skill_path: Path) -> dict:
    """Validate a single skill."""
    skill_md = skill_path / "SKILL.md"
    
    if not skill_md.exists():
        return {"valid": False, "error": "Missing SKILL.md"}
    
    content = skill_md.read_text()
    
    # Check for role declaration
    role_match = re.search(r'\*\*Role\*\*\s*\|\s*(GAY|MASTER|SLAVE)', content)
    if not role_match:
        return {"valid": False, "error": "Missing role declaration"}
    
    role = role_match.group(1)
    trit = {"GAY": 1, "MASTER": 0, "SLAVE": -1}[role]
    
    # Check for color
    color_match = re.search(r'#([0-9A-Fa-f]{6})', content)
    color = color_match.group(0) if color_match else None
    
    return {
        "valid": True,
        "role": role,
        "trit": trit,
        "color": color,
        "name": skill_path.name
    }

def validate_triads(skills: list) -> list:
    """Check GF(3) conservation across skill triads."""
    violations = []
    
    # Group by declared triads
    for i in range(0, len(skills) - 2, 3):
        triad = skills[i:i+3]
        trit_sum = sum(s.get("trit", 0) for s in triad if s.get("valid"))
        
        if trit_sum % 3 != 0:
            violations.append({
                "triad": [s.get("name") for s in triad],
                "sum": trit_sum,
                "violation": True
            })
    
    return violations

def main():
    skills_dir = Path("skills")
    
    if not skills_dir.exists():
        print("No skills directory found")
        return 1
    
    results = []
    for skill_path in sorted(skills_dir.iterdir()):
        if skill_path.is_dir():
            result = validate_skill(skill_path)
            results.append(result)
            status = "✓" if result["valid"] else "✗"
            print(f"{status} {skill_path.name}: {result.get('role', 'unknown')} ({result.get('trit', '?')})")
    
    violations = validate_triads(results)
    
    if violations:
        print(f"\n⚠️  GF(3) Violations: {len(violations)}")
        for v in violations:
            print(f"  - {v['triad']}: sum={v['sum']}")
        return 1
    
    print(f"\n✓ All {len(results)} skills validated, GF(3) conserved")
    return 0

if __name__ == "__main__":
    exit(main())

Test Suite

# tests/test_gf3_validation.py

import pytest

def test_triad_conservation():
    """Verify (+1) + (0) + (-1) = 0."""
    assert (1 + 0 + -1) == 0

def test_role_trit_mapping():
    """Verify role to trit mapping."""
    roles = {"GAY": 1, "MASTER": 0, "SLAVE": -1}
    assert sum(roles.values()) == 0

def test_color_determinism():
    """Verify Gay.jl color is deterministic."""
    # Mock: In production, call actual Gay.jl MCP
    expected = "#4857D5"
    actual = "#4857D5"  # color_at(seed=2025, index=8)
    assert actual == expected

Bisimulation Game Role

As the SLAVE (-1) in the bisimulation game:

1. Attacker move: This skill distinguishes valid from invalid skill structures
2. Constraint function: Enforces GF(3) conservation law
3. Verification: Proves triads sum to zero

Integration with PR Trajectory

This skill is predicted as PR#8 in the plurigrid/asi trajectory:

Triad 3 conservation: (+1) + (-1) + (0) = 0 ✓

XIP Color: #4857D5 Gay.jl Seed: 2025 Gay.jl Index: 8 Role: SLAVE (-1)

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.