Cybernetic Open Game Skill

Compositional game theory for off-chain/on-chain cybernetic feedback loops with GF(3) Nash equilibrium

Trit: 0 (ERGODIC - Coordinator) Color: #26D826 (Green) Status: Production Ready Created: 2025-12-30

Overview

This skill formalizes the Agent-O-Rama ↔ Worldnet ↔ STC cybernetic feedback loop as a compositional open game where:

• Off-chain intelligence (Agent-O-Rama/DuckDB) drives cognition
• On-chain settlement (Secure Ternary Coin/Aptos) provides finality
• Value-conserving bridge (Worldnet) maintains GF(3) balance
• Nash equilibrium = GF(3) conservation across all layers

┌─────────────────────────────────────────────────────────────────────────────┐
│                    CYBERNETIC LOOP AS OPEN GAME                             │
├─────────────────────────────────────────────────────────────────────────────┤
│                                                                             │
│         FORWARD PLAY (Strategies)                                           │
│                                                                             │
│    Intent ──────────▶ Transaction ──────────▶ Settlement                    │
│      X                    Y                      Z                          │
│                                                                             │
│    ┌─────────────┐    ┌─────────────┐    ┌─────────────┐                   │
│    │ Agent-O-Rama│───▶│  Worldnet   │───▶│    STC      │                   │
│    │   (-1)      │    │    (0)      │    │   (+1)      │                   │
│    │   PLAY      │    │   VERIFY    │    │   SETTLE    │                   │
│    └─────────────┘    └─────────────┘    └─────────────┘                   │
│          ▲                  │                  │                            │
│          │                  │                  │                            │
│          │    BACKWARD COPLAY (Utilities)      │                            │
│          │                  │                  ▼                            │
│    ┌─────────────┐    ┌─────────────┐    ┌─────────────┐                   │
│    │ Learning    │◀───│  Reward     │◀───│ Finality    │                   │
│    │   R         │    │    S        │    │    T        │                   │
│    └─────────────┘    └─────────────┘    └─────────────┘                   │
│                                                                             │
│   NASH EQUILIBRIUM = GF(3) Conservation                                     │
│   ∀ strategy s: isEquilibrium(s) ⟺ Σ(trits) ≡ 0 (mod 3)                   │
│                                                                             │
└─────────────────────────────────────────────────────────────────────────────┘

Architecture Layers

Layer 1: Agent-O-Rama (Off-Chain Intelligence)

class AgentORama:
    """
    Off-chain cognition layer.
    High bandwidth, low cost, local-first.
    """
    trit = -1  # MINUS (Validator/Constrainer)

    def __init__(self):
        self.duckdb = DuckDB()  # Local storage
        self.patterns = PatternExtractor()  # Barton surrogate

    def play(self, observation) -> Intent:
        """Forward play: form intent from observation."""
        patterns = self.patterns.extract(observation)
        intent = self.form_intent(patterns)
        return intent

    def coplay(self, intent, reward) -> Learning:
        """Backward coplay: update model from reward."""
        self.patterns.update(intent, reward)
        return Learning(delta=self.patterns.delta)

Layer 2: Worldnet (Value-Conserving Bridge)

class Worldnet:
    """
    Bridge layer with GF(3) verification.
    Immune membrane: Self/Non-Self discrimination.
    """
    trit = 0  # ERGODIC (Coordinator)

    def play(self, intent) -> Transaction:
        """Forward play: verify and transform intent."""
        if not self.verify_gf3(intent):
            raise ConservationViolation("GF(3) violated")
        return Transaction(intent, signature=self.sign(intent))

    def coplay(self, transaction, finality) -> Reward:
        """Backward coplay: propagate finality as reward."""
        return Reward(
            value=finality.value,
            gf3_balanced=self.verify_gf3(finality)
        )

    def verify_gf3(self, data) -> bool:
        """Immune check: is this self (valid) or non-self (invalid)?"""
        return sum(data.trits) % 3 == 0

Layer 3: STC on Aptos (On-Chain Settlement)

class SecureTernaryCoin:
    """
    On-chain settlement layer.
    Low bandwidth, high cost, global consensus.
    """
    trit = +1  # PLUS (Generator/Executor)

    def play(self, transaction) -> Settlement:
        """Forward play: settle on Aptos mainnet."""
        result = self.aptos.submit(transaction)
        return Settlement(
            txn_hash=result.hash,
            finality=result.finality
        )

    def coplay(self, settlement, _) -> Observation:
        """Backward coplay: emit observation for agent."""
        return Observation(
            state=self.aptos.get_state(settlement.txn_hash),
            timestamp=settlement.finality.timestamp
        )

Open Game Formalization

Game Structure

-- Open game type
data OpenGame s t a b = Game
  { play    :: s -> a           -- Forward strategy
  , coplay  :: s -> b -> t      -- Backward utility
  , equilibrium :: s -> Bool    -- Nash condition
  }

-- The full cybernetic game
cyberneticLoop :: OpenGame Intent Settlement Observation Reward
cyberneticLoop = agentGame ; worldnetGame ; stcGame

-- Sequential composition
(;) :: OpenGame a b c d -> OpenGame c d e f -> OpenGame a b e f
g ; h = Game
  { play = play h . play g
  , coplay = \s f -> coplay g s (coplay h (play g s) f)
  , equilibrium = \s -> equilibrium g s && equilibrium h (play g s)
  }

-- Parallel composition
(⊗) :: OpenGame a b c d -> OpenGame e f g h -> OpenGame (a,e) (b,f) (c,g) (d,h)
g ⊗ h = Game
  { play = \(s1, s2) -> (play g s1, play h s2)
  , coplay = \(s1, s2) (t1, t2) -> (coplay g s1 t1, coplay h s2 t2)
  , equilibrium = \(s1, s2) -> equilibrium g s1 && equilibrium h s2
  }

GF(3) as Nash Equilibrium

-- Nash equilibrium iff GF(3) conserved
theorem_gf3_nash :: CyberneticLoop -> Proof
theorem_gf3_nash loop =
  let trits = [trit agentGame, trit worldnetGame, trit stcGame]
      -- [-1, 0, +1]
  in sum trits `mod` 3 == 0  -- Nash condition

-- Compositional equilibrium preservation
theorem_compositional :: Game -> Game -> Proof
theorem_compositional g h =
  equilibrium g && equilibrium h
  ==> equilibrium (g ; h)

Six-Dimensional Review Framework

This skill synthesizes analysis from six cognitive dimensions:

1. Graph Grafting (Structure)

# Architecture as graft complex
arch = GraftComplex(UInt64(1069))
graft!(arch, agent_o_rama, :none, String[])
graft!(arch, worldnet, :agent_o_rama, ["value_conservation"])
graft!(arch, stc, :worldnet, ["aptos_mainnet"])
verify_gf3(arch)  # → (conserved=true, sum=0)

2. ACSets (Categorical Schema)

@present SchCyberneticGame(FreeSchema) begin
    Agent::Ob; Bridge::Ob; Chain::Ob
    Intent::Ob; Transaction::Ob; Settlement::Ob

    emit::Hom(Agent, Intent)
    verify::Hom(Intent, Bridge)
    submit::Hom(Bridge, Transaction)
    settle::Hom(Transaction, Chain)
    observe::Hom(Chain, Agent)  # Feedback!

    Trit::AttrType
    agent_trit::Attr(Agent, Trit)     # -1
    bridge_trit::Attr(Bridge, Trit)   # 0
    chain_trit::Attr(Chain, Trit)     # +1
end

3. Bisimulation Game (Verification)

ROUND 1: Attacker submits invalid intent
         Defender: "Rejected - GF(3) violated"
         Arbiter: BLOCKED ✓

ROUND 2: Attacker attempts TOCTOU attack
         Defender: "Signature invalidated by post-mod"
         Arbiter: BLOCKED ✓

VERDICT: Bisimilar to secure spec (depth 3)

4. BlackHat-Go / Carlini (Security)

5. Cybernetic Immune (Self/Non-Self)

SELF (Reafference):           NON-SELF (Exafference):
• Valid GF(3) intents         • Conservation violation
• Signed by known agent       • Unknown signature
• Within rate limits          • Anomalous patterns

WORLDNET = Immune Membrane (MHC Presentation)

6. ALife (Emergence)

METABOLISM: Value flows through layers
REPRODUCTION: Successful patterns replicate
HOMEOSTASIS: GF(3) conservation maintains stability
COOPERATION: TIT-FOR-TAT strategies emerge

Condensed Mathematics Connection

module CondensedCyberneticGame
  # Each layer has a liquid parameter encoding bandwidth/cost
  LAYERS = {
    agent_o_rama: { trit: -1, liquid_r: 0.3 },  # High BW, local
    worldnet:     { trit:  0, liquid_r: 0.5 },  # Medium
    stc:          { trit: +1, liquid_r: 0.9 }   # Low BW, global (solid)
  }

  # Liquid → Solid transition = Off-chain → On-chain
  def self.solidify(intent, r_threshold: 0.9)
    # As r → 1, we approach on-chain settlement
    current_r = intent.layer.liquid_r
    if current_r >= r_threshold
      submit_to_aptos(intent)
    else
      keep_offchain(intent)
    end
  end
end

Mutual Awareness Framework

Open games provide semantics for skill mutual awareness:

-- Each skill is a game
skillGame :: Skill -> OpenGame Context Result Query Response

-- Mutual awareness = game composition
aware :: Skill -> Skill -> OpenGame
aware s1 s2 = skillGame s1 ; skillGame s2

-- Awareness network = parallel composition
awarenessNetwork :: [Skill] -> OpenGame
awarenessNetwork = foldr1 (⊗) . map skillGame

-- Equilibrium = mutual consistency
isAware :: [Skill] -> Bool
isAware skills = equilibrium (awarenessNetwork skills)

DuckDB Activity Aggregation

-- Skills become mutually aware via activity correlation
CREATE TABLE skill_awareness (
    from_skill VARCHAR,
    to_skill VARCHAR,
    awareness_type VARCHAR,  -- 'neighbor', 'ref', 'cooccur', 'game'
    weight FLOAT,
    game_equilibrium BOOLEAN,  -- Nash equilibrium?
    PRIMARY KEY (from_skill, to_skill, awareness_type)
);

-- Open game equilibrium check
CREATE VIEW game_equilibrium AS
SELECT
    from_skill, to_skill,
    SUM(trit) % 3 = 0 as nash_equilibrium
FROM skill_awareness sa
JOIN skill_trits st ON sa.from_skill = st.skill
GROUP BY from_skill, to_skill;

GF(3) Triads

# Core Architecture Triads
agent-o-rama (-1) ⊗ cybernetic-open-game (0) ⊗ stc (+1) = 0 ✓
bisimulation-game (-1) ⊗ open-games (0) ⊗ alife (+1) = 0 ✓
blackhat-go (-1) ⊗ cybernetic-open-game (0) ⊗ cooperation (+1) = 0 ✓
condensed-stacks (-1) ⊗ cybernetic-open-game (0) ⊗ emergence (+1) = 0 ✓

# Verification Triads
carlini-attack (-1) ⊗ immune-membrane (0) ⊗ homeostasis (+1) = 0 ✓
temporal-coalgebra (-1) ⊗ open-games (0) ⊗ free-monad-gen (+1) = 0 ✓
sheaf-cohomology (-1) ⊗ graph-grafting (0) ⊗ nash-equilibrium (+1) = 0 ✓

Integrated Skills

This skill synthesizes knowledge from:

Commands

# Verify GF(3) conservation
just cybernetic-verify

# Run bisimulation game
just cybernetic-bisim depth=5

# Check Nash equilibrium
just cybernetic-nash

# Analyze attack surface
just cybernetic-security

# Run immune classification
just cybernetic-immune intent.json

# Export to DuckDB
just cybernetic-export

# Visualize game network
just cybernetic-viz

References

• Ghani, Hedges, et al. "Compositional Game Theory"
• Capucci & Gavranović, "Actegories for Open Games"
• Scholze & Clausen, "Condensed Mathematics"
• Varela, "Principles of Biological Autonomy"
• Axelrod, "Evolution of Cooperation"
• Carlini, "Evaluating Adversarial Robustness"

See Also

• `open-games` - Core open game theory
• `bisimulation-game` - Verification games
• `agent-o-rama` - Cognitive surrogate
• `graph-grafting` - Compositional structure
• `cybernetic-immune` - Immune discrimination
• `duckdb-ies` - Activity aggregation

Skill Name: cybernetic-open-game Type: Architecture / Game Theory / Systems Integration Trit: 0 (ERGODIC - coordinator) GF(3): Conserved via Nash equilibrium Thread: 2025-12-30 cognitive superposition review

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

• game-theory: 21 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.