Trialectical Coin

Trialectical Coin Skill

Extends TeglonLabs/coin-flip-mcp with GF(3) ternary decision logic.

Source

• Repository: TeglonLabs/coin-flip-mcp
• Trit: 0 (ERGODIC - meta-decision coordinator)

Overview

The trialectical coin replaces binary true/false with ternary −/○/+ outcomes:

GF(3) Conservation

Every trialectical flip maintains conservation:

• Single flip: outcome ∈ {−1, 0, +1}
• Triple flip: Σ(outcomes) ≡ 0 (mod 3) when balanced
• Meta-decisions use the ERGODIC outcome to defer to oracle

MCP Tools

trialectical_flip

Flip a 3-sided coin with GF(3) semantics.

{
  "name": "trialectical_flip",
  "arguments": {
    "seed": 1069,
    "context": "Should I generate or validate?"
  }
}

Returns:

{
  "outcome": "+",
  "trit": 1,
  "meaning": "PLUS: Generate, create, affirm",
  "color": "#E67F86",
  "gf3_balanced": true
}

trialectical_batch

Flip N coins, ensuring GF(3) conservation.

{
  "name": "trialectical_batch",
  "arguments": {
    "count": 6,
    "seed": 1069
  }
}

Returns:

{
  "outcomes": ["+", "−", "○", "+", "−", "○"],
  "trits": [1, -1, 0, 1, -1, 0],
  "sum": 0,
  "conserved": true
}

trialectical_oracle

Meta-decision: flip until reaching consensus (3 same outcomes).

{
  "name": "trialectical_oracle",
  "arguments": {
    "question": "Which skill to invoke?",
    "options": ["narya-proofs", "gay-mcp", "bisimulation-game"]
  }
}

Ising State Model

The trialectical coin implements an Ising model where:

• Spins: σᵢ ∈ {−1, 0, +1}
• Energy: E = −J Σ σᵢσⱼ (ferromagnetic coupling)
• Temperature: τ controls randomness vs determinism

At τ=0: Deterministic (same seed → same outcome) At τ→∞: Maximum entropy (equal probability) At τ=τ*: Critical point (BKT transition)

Usage Patterns

Decision Branching

flip = trialectical_flip()
if flip.trit == 1:
    # PLUS: Generate new content
elif flip.trit == -1:
    # MINUS: Validate existing content
else:
    # ERGODIC: Defer to oracle or user

Triadic Task Assignment

flips = trialectical_batch(3)
# Assign to subagents based on trit:
# flips[1] → MINUS validator
# flips[2] → ERGODIC coordinator  
# flips[3] → PLUS generator

GF(3)-Balanced Exploration

# Ensure sum of decisions = 0 (mod 3)
while not conserved:
    flips = trialectical_batch(n)
    if sum(flips.trits) % 3 == 0:
        conserved = true

Random.org Integration

For true randomness (not pseudo-random):

1. Query random.org for integer in [0, 2]
2. Map: 0 → −1 (Tails), 1 → 0 (_), 2 → +1 (Heads)
3. Cache for offline use with Gay.jl fallback

Skill Correspondences

Installation

# Clone TeglonLabs repo
git clone https://github.com/TeglonLabs/coin-flip-mcp
cd coin-flip-mcp
npm install
npm run build

# Add to Claude config
# ~/.config/claude/claude_desktop_config.json

Configuration

{
  "mcpServers": {
    "trialectical-coin": {
      "command": "node",
      "args": ["/path/to/coin-flip-mcp/build/index.js"],
      "env": {
        "RANDOM_ORG_API_KEY": "your-key-here"
      }
    }
  }
}

End-of-Skill Interface

Integration with Gay.jl

using Gay

# Seed from gay_seed for reproducibility
gay_seed!(1069)

# Flip using Gay.jl color-to-trit mapping
color = gay_color_at(42)
trit = gay_trit_at(42)  # Derived from hue

outcome = trit == 1 ? "Heads" : trit == -1 ? "Tails" : "_"

References

• AIP-41 - Aptos on-chain randomness
• Gay.jl - Deterministic color generation
• Ising Model - Statistical mechanics basis
• GF(3) - Field with 3 elements

Autopoietic Marginalia

The interaction IS the skill improving itself.

Every use of this skill is an opportunity for worlding:

• MEMORY (-1): Record what was learned
• REMEMBERING (0): Connect patterns to other skills
• WORLDING (+1): Evolve the skill based on use

Add Interaction Exemplars here as the skill is used.