Open Games Plurigrid Skill

Open Games Plurigrid Skill

World-to-world interactions via compositional game theory with GF(3) energy consistency

Trit: 0 (ERGODIC)
Color: #26D826 (Green)
Synergy: 0.59 (plurigrid ↔ jules-hedges)

Overview

This skill unifies:

• open-games: Play/​CoPlay bidirectional optics
• ramanujan-expander: Spectral gap λ₂ ≤ 2√(d-1)
• glass-bead-game: Triangle inequality d(W₁,W₃) ≤ d(W₁,W₂) + d(W₂,W₃)

Core Insight

Play-Co-Play interactions are energy-consistent open games:

        ┌───────────────┐
   X ──→│               │──→ Y
        │  δ-Key Update │
   R ←──│               │←── S
        └───────────────┘

  Play (+1):   forward pass, propose action, derive delta
  CoPlay (-1): backward pass, adjust world, verify proof
  Bridge (0):  equilibrium, combine deltas, coordinate
  
  GF(3): (+1) + (-1) + (0) = 0 ✓ (energy conserved)

Discovered Ecosystem

Core Contributors (Git Log)

Repositories

Key Bridge Person

clayrat (Alex Gryzlov) @ IMDEA Software

• Stars BOTH open-games AND Catlab.jl
• Repos: dialectica, coherence-spaces, comonads
• The cobordism between compositional game theory ↔ applied category theory

bgavran (Bruno Gavranović) - Categorical Deep Learning

• 1459 stars on Category_Theory_Machine_Learning list
• Lens_Resources (56 stars) - optics theory
• Compositional_Deep_Learning (151 stars)

World Hopping

Distance Metric

def world_distance(w1, w2, dr):
    # Hamming distance on fingerprints (being)
    being_diff = bin(fp1 ^ fp2).count('1') / 64.0
    # Epoch difference (temporal)
    epoch_diff = abs(w1.epoch - w2.epoch) / 100.0
    # Domain crossing
    domain_diff = 0.0 if w1.domain == w2.domain else 0.3
    
    return sqrt(being_diff² + epoch_diff² + domain_diff²)

Triangle Inequality

For any hop W₁ → W₃ via W₂:

d(W₁, W₃) ≤ d(W₁, W₂) + d(W₂, W₃)

Elegance = direct_distance / path_distance (higher = better)

Spectral Analysis

Ramanujan Bound

For d-regular expander graph:

λ₂ ≤ 2√(d-1)

Observed: λ₁ = 0.053 after growing, bound = 2.14

Growing the Expander

Add edges that maximize spectral gap:

1. Prefer cross-cluster connections
2. Maintain GF(3) balance
3. Respect domain boundaries

Files

• open_games_expander.py - Graph discovery
• play_coplay_protocol.py - Delta update protocol
• plurigrid_world_hop.py - World hopping engine
• delta_key_update.py - Original delta protocol
• gay_loot.py - Synergy-based loot tables

GF(3) Triads

temporal-coalgebra (-1) ⊗ open-games (0) ⊗ free-monad-gen (+1) = 0 ✓
sheaf-cohomology (-1) ⊗ open-games (0) ⊗ topos-generate (+1) = 0 ✓
ramanujan-expander (-1) ⊗ glass-bead-game (0) ⊗ world-hopping (+1) = 0 ✓

Commands

# Run world hopping
python3 plurigrid_world_hop.py

# Play-CoPlay protocol
python3 play_coplay_protocol.py

# Expander graph discovery
python3 open_games_expander.py

# Delta key update demo
python3 delta_key_update.py

Related Skills

• open-games - Core game theory
• ramanujan-expander - Spectral bounds
• glass-bead-game - World hopping
• delta-key-update - Key rotation protocol
• gay-mcp - Color fingerprints
• bisimulation-game - Observational equivalence

Para(Optic) atlas

Part of: para-mensch-commons.