cybernetic-open-game

# 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 | Component | Trit | Role | Bandwidth | Cost |
|-------|-----------|------|------|-----------|------|
| 1 | Agent-O-Rama | -1 | Cognition | High | Low |
| 2 | Worldnet | 0 | Bridge/Verification | Medium | Medium |
| 3 | STC (Aptos) | +1 | Settlement/Consensus | Low | High |

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

```python
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)

```python
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)

```python
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

```haskell
-- 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

```haskell
-- 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)

```julia
# 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)

```julia
@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)

| Attack Vector | Risk | Mitigation |
|---------------|------|------------|
| Intent Injection | 8/10 | Prompt validation |
| Trit Manipulation | 9/10 | Cryptographic signing |
| Replay Attack | 7/10 | Nonce + timestamp |
| TOCTOU | 8/10 | Atomic commit-sign |
| GF(3) Bypass | 10/10 | Formal verification |

### 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

```ruby
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:

```haskell
-- 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

```sql
-- 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:

| Skill | Trit | Contribution |
|-------|------|--------------|
| `_integrated` | 0 | ASI skill orchestration |
| `bisimulation-game` | -1 | Verification protocol |
| `acsets` | 0 | Categorical schema |
| `cognitive-superposition` | 0 | Multi-perspective analysis |
| `blackhat-go` | -1 | Security analysis |
| `cybernetic-immune` | 0 | Self/Non-Self discrimination |
| `alife` | +1 | Emergence patterns |
| `condensed-analytic-stacks` | -1 | Topology/bandwidth |
| `graph-grafting` | 0 | Compositional structure |
| `open-games` | 0 | Game-theoretic semantics |
| `reflow` | 0 | Information transformation |
| `duckdb-ies` | +1 | Activity aggregation |

## Commands

```bash
# 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`](../open-games/SKILL.md) - Core open game theory
- [`bisimulation-game`](../bisimulation-game/SKILL.md) - Verification games
- [`agent-o-rama`](../agent-o-rama/SKILL.md) - Cognitive surrogate
- [`graph-grafting`](../graph-grafting/SKILL.md) - Compositional structure
- [`cybernetic-immune`](../cybernetic-immune/SKILL.md) - Immune discrimination
- [`duckdb-ies`](../duckdb-ies/SKILL.md) - 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.