--- name: unwiring-arena description: Play/Coplay arena theory for autopoietic closure with GF(3) conservation version: 1.0.0 --- # Unwiring Arena Skill **Status**: ✅ Production Ready **Trit**: 0 (ERGODIC - balanced flow) **Principle**: Play/Coplay autopoietic closure with GF(3) conservation **Source**: plurigrid/UnwiringDiagrams.jl#1 + Capucci et al. Arena Theory --- ## Overview **Unwiring Arena** unifies three categorical patterns: 1. **Wiring Diagrams** (AlgebraicJulia/Catlab) - Compositional system construction 2. **Unwiring Rules** (GayUncommonsSimulator) - Learning through constraint release 3. **Arena Protocol** (Plurigrid Amelia v0.4) - Play/Coplay bidirectional channels ``` ┌─ Play Channel (Outbound) ─────────────────┐ │ Local arena mutations → NATS broadcast │ │ Strategy profiles → Action selection │ └────────────────────────────────────────────┘ ↕ (Autopoietic closure) ┌─ Coplay Channel (Inbound) ───────────────┐ │ Peer arena submissions → Reconciliation │ │ Rewards/feedback → Identity update │ └────────────────────────────────────────────┘ ``` ## Mathematical Foundation ### Arenas as Parametrised Lenses From Capucci, Ghani, Ledent, Forsberg: ``` Arena A_G : Lens_{(Ω,℧)}(X,S)(Y,R) where: Ω = Π_{p∈P} Ωₚ (strategy profiles) ℧ = Π_{p∈P} ℧ₚ (reward vectors) X, Y = states S, R = costates (feedback) ``` **Play**: `play_A : Ω × X → Y` (forward pass) **Coplay**: `coplay_A : Ω × X × R → ℧ × S` (backward pass with feedback) ### Unwiring = Learning Through Constraint Release ```julia struct UnwiringRule source_gf3::Int # Source polarity {-1, 0, +1} target_gf3::Int # Target polarity learning_rate::Float64 # How fast to unwire threshold::Float64 # Discrepancy threshold to trigger end # Unwiring shifts internal toward external (learning) function apply_unwiring(rule, internal, external) α = rule.learning_rate return (1-α) * internal + α * external end ``` ### GF(3) Tripartite Channels ``` MINUS (-1) : Constraint verification (coplay focus) ERGODIC (0) : Balance/coordination (arena equilibrium) PLUS (+1) : Generative exploration (play focus) Conservation: sum(trits) ≡ 0 (mod 3) per triplet ``` ## Arena Protocol v0.4 (Amelia) ### Schema (JSONSchema v7) ```json { "$schema": "http://json-schema.org/draft-07/schema#", "type": "object", "properties": { "arena_id": { "type": "string", "format": "uuid" }, "seed": { "type": "integer", "description": "SplitMix64 seed" }, "players": { "type": "array", "items": { "type": "object", "properties": { "id": { "type": "string" }, "strategy_space": { "type": "array" }, "gf3_polarity": { "enum": [-1, 0, 1] } } } }, "play_channel": { "type": "object", "properties": { "topic": { "type": "string", "pattern": "^plurigrid\\.arena\\.play\\." }, "mutations": { "type": "array" } } }, "coplay_channel": { "type": "object", "properties": { "topic": { "type": "string", "pattern": "^plurigrid\\.arena\\.coplay\\." }, "reconciliations": { "type": "array" } } } } } ``` ### NATS Topics ``` plurigrid.arena.play.{agent}.{focus} # Outbound mutations plurigrid.arena.coplay.{agent}.{focus} # Inbound feedback plurigrid.arena.peers.> # Peer discovery plurigrid.arena.reconcile # Conflict resolution ``` ## Julia Implementation ### Core Types ```julia using Catlab.WiringDiagrams using Catlab.Programs # Arena as parametrised lens struct Arena{Ω, U, X, S, Y, R} play::Function # Ω × X → Y coplay::Function # Ω × X × R → U × S strategies::Ω rewards::U end # Unwiring rule for learning struct UnwiringRule source_gf3::Int target_gf3::Int learning_rate::Float64 threshold::Float64 end # Agent with reafferent identity mutable struct ArenaAgent id::UInt64 internal_state::Vector{Float64} external_observation::Vector{Float64} discrepancy::Float64 gf3::Int unwiring_count::Int current_rule::Int end ``` ### Compositional Arena Construction ```julia using Catlab.WiringDiagrams: WiringDiagram, add_box!, add_wire! """ Build arena wiring diagram from player specifications. """ function build_arena_diagram(players::Vector{Tuple{Symbol, Int}}) d = WiringDiagram([:X], [:Y]) for (name, arity) in players box = add_box!(d, Box(name, fill(:strategy, arity), [:action])) end # Wire sequentially (can be customized) for i in 1:(length(players)-1) add_wire!(d, (i, 1) => (i+1, 1)) end return d end """ Unwire a connection (remove constraint, enable learning). """ function unwire!(d::WiringDiagram, wire_id::Int) # Remove wire - allows independent evolution delete_wire!(d, wire_id) end """ Apply unwiring rule based on discrepancy. """ function apply_unwiring!(agent::ArenaAgent, rules::Vector{UnwiringRule}) rule = rules[agent.current_rule] if agent.discrepancy > rule.threshold && agent.gf3 == rule.source_gf3 α = rule.learning_rate agent.internal_state = (1-α) .* agent.internal_state .+ α .* agent.external_observation agent.unwiring_count += 1 agent.current_rule = mod1(agent.current_rule + 1, length(rules)) end update_identity!(agent) end ``` ### Play/Coplay Implementation ```julia """ Play: Forward pass through arena. """ function arena_play(arena::Arena, strategy::Ω, state::X) where {Ω, X} arena.play(strategy, state) end """ Coplay: Backward pass with reward feedback. """ function arena_coplay(arena::Arena, strategy::Ω, state::X, reward::R) where {Ω, X, R} arena.coplay(strategy, state, reward) end """ Autopoietic closure: play ∘ coplay cycle. """ function arena_cycle!(agent::ArenaAgent, arena::Arena, external_reward) # Play: agent acts action = arena_play(arena, agent.internal_state, agent.external_observation) # Coplay: receive feedback, update internal state utility, new_costate = arena_coplay(arena, agent.internal_state, agent.external_observation, external_reward) # Update discrepancy (reafference) agent.external_observation = new_costate agent.discrepancy = norm(agent.internal_state - agent.external_observation) agent.gf3 = Int(hash(agent.internal_state) % 3) - 1 return action, utility end ``` ## Python Implementation ```python """ unwiring_arena.py - Arena Protocol with Unwiring Rules """ from dataclasses import dataclass, field from typing import List, Callable, Tuple, Dict, Any import numpy as np # SplitMix64 constants GOLDEN = 0x9E3779B97F4A7C15 MASK64 = (1 << 64) - 1 @dataclass class UnwiringRule: """How identity transforms through learning.""" source_gf3: int # {-1, 0, +1} target_gf3: int learning_rate: float threshold: float UNWIRING_RULES = [ UnwiringRule(0, 1, 0.1, 0.2), # Fresh→Aging: slow learning UnwiringRule(1, -1, 0.3, 0.4), # Aging→Stale: medium learning UnwiringRule(-1, 0, 0.5, 0.6), # Stale→Fresh: fast learning (reset) UnwiringRule(0, -1, 0.05, 0.8), # Fresh→Stale: rare leap ] @dataclass class ArenaAgent: """Agent with reafferent identity.""" id: int internal_state: np.ndarray external_observation: np.ndarray discrepancy: float = 0.0 gf3: int = 0 unwiring_count: int = 0 current_rule: int = 0 def update_identity(self): """Recompute identity from internal/external discrepancy.""" self.discrepancy = np.linalg.norm( self.internal_state - self.external_observation ) h = hash(self.internal_state.tobytes()) self.gf3 = (h % 3) - 1 @dataclass class Arena: """Parametrised lens for game-theoretic interaction.""" play: Callable # (strategy, state) -> action coplay: Callable # (strategy, state, reward) -> (utility, costate) def cycle(self, agent: ArenaAgent, external_reward: float) -> Tuple[Any, float]: """Execute play/coplay autopoietic cycle.""" # Play: forward pass action = self.play(agent.internal_state, agent.external_observation) # Coplay: backward pass with feedback utility, new_costate = self.coplay( agent.internal_state, agent.external_observation, external_reward ) # Update agent's external observation agent.external_observation = new_costate agent.update_identity() return action, utility def apply_unwiring(agent: ArenaAgent, rules: List[UnwiringRule] = UNWIRING_RULES): """Apply unwiring rule to shift internal toward external.""" rule = rules[agent.current_rule] if agent.discrepancy > rule.threshold and agent.gf3 == rule.source_gf3: α = rule.learning_rate agent.internal_state = ( (1 - α) * agent.internal_state + α * agent.external_observation ) agent.unwiring_count += 1 agent.current_rule = (agent.current_rule + 1) % len(rules) agent.update_identity() ``` ## NATS Integration ```python import nats import json NATS_SERVER = "nats://nonlocal.info:4222" async def publish_arena_play(nc, agent_id: str, focus: str, mutation: dict): """Publish play channel mutation.""" topic = f"plurigrid.arena.play.{agent_id}.{focus}" await nc.publish(topic, json.dumps(mutation).encode()) async def subscribe_arena_coplay(nc, agent_id: str, handler): """Subscribe to coplay channel feedback.""" topic = f"plurigrid.arena.coplay.{agent_id}.>" await nc.subscribe(topic, cb=handler) async def broadcast_reconciliation(nc, arena_state: dict): """Broadcast arena state for reconciliation.""" await nc.publish("plurigrid.arena.reconcile", json.dumps(arena_state).encode()) ``` ## Integration with GayUncommonsSimulator ```julia # From GayUncommonsSimulator.jl - the key insight # Wiring = constraint (commons pooling) # Unwiring = freedom (uncommons differentiation) const UNWIRING_RULES = [ UnwiringRule(0, 1, 0.1, 0.2), # Fresh→Aging UnwiringRule(1, 2, 0.3, 0.4), # Aging→Stale UnwiringRule(2, 0, 0.5, 0.6), # Stale→Fresh (reset cycle) ] # Identity IS discrepancy struct ReafferentID internal::RGB external::RGB discrepancy::Float64 hash::UInt64 gf3::Int end # ID = hash(internal ⊻ external) function ReafferentID(internal::RGB, external::RGB) disc = abs(spectrum_t(internal) - spectrum_t(external)) id_hash = mix(to_hash(internal) ⊻ to_hash(external)) gf3 = Int(id_hash % 3) ReafferentID(internal, external, disc, id_hash, gf3) end ``` ## Chronoscope Video Integration The Arena Protocol includes video orchestration for 69 demonstrations: ```julia # 7 categories × ~10 videos each CHRONOSCOPE_CATEGORIES = [ :categorical_ops, # Operad operations :quantum, # ZX calculus :hypergraph, # Hypergraph rewriting :ducklake, # Time-travel queries :embodied_gradualism, # Learning dynamics :ontology, # Schema evolution :arena_protocol, # This very protocol ] # Deterministic scheduling with seed 1069 schedule_videos(seed=1069, categories=CHRONOSCOPE_CATEGORIES) ``` ## Commands ```bash # Julia - run arena simulation julia -e "include(\"GayUncommonsSimulator.jl\"); simulate_wev(n_steps=100)" # Python - NATS arena client uv run --with nats-py arena_client.py # Catlab wiring diagram julia -e "using Catlab.WiringDiagrams; d = WiringDiagram([:X], [:Y])" ``` ## Key Patterns | Pattern | Wiring | Unwiring | |---------|--------|----------| | **Semantics** | Constraint | Freedom | | **Direction** | Composition | Decomposition | | **Learning** | Fixed relationship | Adaptive relationship | | **GF3 Flow** | -1 (MINUS) | +1 (PLUS) | ## See Also - [GayUncommonsSimulator.jl](file:///Users/bob/ies/GayUncommonsSimulator.jl) - Full implementation - [plurigrid/UnwiringDiagrams.jl#1](https://github.com/plurigrid/UnwiringDiagrams.jl/pull/1) - Arena Protocol PR - [Capucci et al.](file:///Users/bob/ies/paper_extracts/mathpix_snips/obsidian/math/M.%20Capucci,%20N.%20Ghani,%20J.%20Ledent,%20F.%20Nordvall%20Forsberg.md) - Arenas with players - [gay-mcp](file:///Users/bob/.claude/skills/gay-mcp/SKILL.md) - Deterministic colors - [triad-interleave](file:///Users/bob/.claude/skills/triad-interleave/SKILL.md) - GF(3) scheduling ## Thread History (Extracted Patterns) ### Core Arena/Games Threads | Thread | Messages | Key Pattern | |--------|----------|-------------| | [GayUncommonsSimulator with reafference](https://ampcode.com/threads/T-019b22bf-11fd-73ad-a61a-e7c40373fd49) | 61 | ID = hash(internal ⊻ external), WEV extraction | | [GAY protocol stack + open games](https://ampcode.com/threads/T-019b22cd-ef86-77eb-a285-8dbe2312558f) | 39 | 3-coloring guarantees, Hedges open games | | [DiscoHy operadic framework](https://ampcode.com/threads/T-019b44e5-39bb-7251-a8fa-5d1efaa5dafa) | 55 | ParaLens, Hedges equilibria, operads | | [Random walk spectral gap](https://ampcode.com/threads/T-019b43de-907c-7008-a545-57e8ff698498) | 104 | Open-games + active-inference GF(3) balance | ### Wiring/Category Threads | Thread | Messages | Key Pattern | |--------|----------|-------------| | [Load acset skill](https://ampcode.com/threads/T-019b43a6-11f7-77a8-a9cd-ea618e457b70) | 88 | C-set as functor, DPO rewriting | | [Formalize skills with interaction entropy](https://ampcode.com/threads/T-019b44d2-05db-72e2-9a4a-23b7e1085b25) | 61 | DisCoPy boxes, ACSet integration | | [Install plurigrid ASI skills](https://ampcode.com/threads/T-019b44c6-bfa3-7371-b238-023eb308d12b) | 124 | SchOperadNetwork, morphism traversal | | [P2P file exchange + skills](https://ampcode.com/threads/T-019b4464-a75b-714e-9f73-e4e1dfd82ab7) | 162 | DiscoPy operads, 1069 subscales | ### WEV/Value Extraction Threads | Thread | Messages | Key Pattern | |--------|----------|-------------| | [Comprehensive GAY protocol](https://ampcode.com/threads/T-019b22d0-c29f-72e7-a229-57a99774d3fa) | 160 | 3-match guarantee, pigeonhole bound | | [Find recent fork threads](https://ampcode.com/threads/T-019b22aa-e7ef-779c-9246-9977c1c6a9ae) | 150 | WEV formula, colored operad schema | | [Self-reflexive triadic subagent](https://ampcode.com/threads/T-019b233a-7ddf-734c-b117-2dd65ae97a80) | 141 | Tripartite network, GF(3) conservation | ### Key Extracted Formulas ```julia # WEV (World Extractable Value) WEV(agent, world) = base_value × staleness_mult × scarcity_mult where: base_value = agent.discrepancy × 10.0 staleness_mult = 1.0 + Σ(days_since_asked) × 0.1 scarcity_mult = 1.0 + (1.0 - occupancy[color] / total) # 3-Coloring Guarantees ∀ i: color[i] ≠ color[i+1] # Adjacent Distinct ∀ window of 3: {R, Y, B} all present # 3-Match Property # Reafferent Identity ID = hash(internal_color ⊻ external_color) discrepancy = |spectrum_t(internal) - spectrum_t(external)| ``` ### ACSet Schema for Arena Protocol ```julia @present SchArenaProtocol(FreeSchema) begin Agent::Ob Arena::Ob Channel::Ob Message::Ob agent_arena::Hom(Agent, Arena) msg_channel::Hom(Message, Channel) channel_arena::Hom(Channel, Arena) # Attributes Name::AttrType Trit::AttrType Seed::AttrType agent_name::Attr(Agent, Name) agent_trit::Attr(Agent, Trit) arena_seed::Attr(Arena, Seed) channel_type::Attr(Channel, Name) # "play" or "coplay" end ``` --- **Skill Name**: unwiring-arena **Type**: Compositional Game Theory / Learning **Trit**: 0 (ERGODIC) **GF(3)**: Play/Coplay closure maintains conservation **SPI**: Deterministic arena fingerprints via seed chain **Threads**: 15+ related threads with 1400+ combined messages ## 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: 10. Adventure Game Example **Concepts**: autonomous agent, game, synthesis ### GF(3) Balanced Triad ``` unwiring-arena (−) + SDF.Ch10 (+) + [balancer] (○) = 0 ``` **Skill Trit**: -1 (MINUS - verification) ### Secondary Chapters - Ch8: Degeneracy - Ch3: Variations on an Arithmetic Theme - Ch4: Pattern Matching - Ch6: Layering - Ch7: Propagators ### Connection Pattern Adventure games synthesize techniques. This skill integrates multiple patterns. ## 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.