Universal CapTP Derivation Skill

Traces capability derivation chains through 3×3 expert poset with GF(3) conservation.

Core Principle

A capability IS the (seed, index) tuple. Expertise IS the trit alignment.

3×3 Expert Poset Structure

Layer	Validator (-1)	Coordinator (0)	Generator (+1)	Domain
1	Jules Hedges	John Baez	Fabrizio Genovese	categorical-games
2	David Egolf	Mike Shulman	sarahzrf	sheaf-theory
3	Betweenness	Mantissa	ModalNoah	neural-categorical

Derivation Chains

capability(-1) → vat(0) → invoke(+1)     [spawn-lifecycle]
membrane(-1)   → netlayer(0) → lambda(+1) [invoke-path]
handoff(-1)    → syrup(0) → sturdy_ref(+1) [transfer-path]
revoke(-1)     → promise(0) → goblin(+1)   [revoke-path]

Superposition States

Each capability exists in superposition across expert domains:

• generator-dominant: Red channel highest (trit=+1)
• coordinator-dominant: Green channel highest (trit=0)
• validator-dominant: Blue channel highest (trit=-1)
• superposed: Balanced RGB

GF(3) Conservation

Term Σ + Expert Σ ≡ 0 (mod 3)

Current state: COHERENT (residue = 0)

Commands

# Trace derivation chains
python3 -c "import duckdb; c=duckdb.connect('culture_evolution.duckdb'); print(c.execute('SELECT * FROM derivation_chains').df())"

# View expert triads
python3 -c "import duckdb; c=duckdb.connect('culture_evolution.duckdb'); print(c.execute('SELECT * FROM expert_triads').df())"

# Query NOW superposition
python3 -c "import duckdb; c=duckdb.connect('culture_evolution.duckdb'); print(c.execute('SELECT * FROM now_superposition').df())"

# Universal skill derivation
python3 -c "import duckdb; c=duckdb.connect('culture_evolution.duckdb'); print(c.execute('SELECT * FROM universal_skill_derivation LIMIT 20').df())"

Integration Points

Database	Table/View	Purpose
culture_evolution.duckdb	derivation_chains	CapTP flow traces
culture_evolution.duckdb	expert_triads	3×3 expert poset
culture_evolution.duckdb	capability_poset	Lattice structure
culture_evolution.duckdb	now_superposition	Current state
culture_evolution.duckdb	universal_skill_derivation	Full matrix
gh_interactome.duckdb	captp_third_party_contributions	Contributor mapping

Key Files

• lib/culture_alter.rb - Stigmergic evolution
• lib/color_capability.rb - CapTP goblin swarm
• lib/interactome_open_games.rb - Expert triads
• scripts/culture_games.rb - Culture × Open Games

Trit Assignments

Trit	Role	Color Channel	Experts
-1	Validator	Blue	Hedges, Egolf, Betweenness
0	Coordinator	Green	Baez, Shulman, Mantissa
+1	Generator	Red	Genovese, sarahzrf, ModalNoah

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: 6. Layering

Concepts: layered data, metadata, provenance, units

GF(3) Balanced Triad

universal-captp-derivation (+) + SDF.Ch6 (+) + [balancer] (+) = 0

Skill Trit: 1 (PLUS - generation)

Secondary Chapters

• Ch10: Adventure Game Example
• Ch1: Flexibility through Abstraction

Connection Pattern

Layering adds metadata. This skill tracks provenance or annotations.

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.