--- name: structural-rewilding description: "Homotopical approach to Artificial Life where 'life' is the topology of changes (diffs). Three orthogonal directions: Behavioral (→), Structural (↓), Bridge (↘) with Narya interaction-time verification." version: 1.0.0 --- # Structural Rewilding: Homotopical Artificial Life > *"Life is not just the state of the system, but the topology of the changes (diffs) it can undergo."* > — zubyul synthesis ## Overview **Structural Rewilding** applies homotopy type theory to Artificial Life, treating organisms as **morphisms between states** rather than states themselves. The key insight: verification happens at **interaction time** via Narya bridge types, not static self-verification. ## The Three Orthogonal Vectors of Change ``` STRUCTURAL (↓) Type/Form Diff │ │ δS: Diff Type A B │ ▼ ┌─────────────────────────────┐ │ │ │ BEHAVIORAL (→) │ │ State/Function Diff │──────────────────────▶ │ δB: path within type │ time evolution │ │ └─────────────────────────────┘ │ │ BRIDGE (↘) │ Coherence Diff │ δC: 2-cell verifying δS preserves δB ▼ ``` | Vector | Symbol | Meaning | Narya Term | |--------|--------|---------|------------| | **Horizontal** | δB | Behavioral/State Diff | Path within type | | **Vertical** | δS | Structural/Type Diff | `Diff Type A B` | | **Diagonal** | δC | Bridge/Coherence Diff | 2-cell, "diff of diffs" | ## Interaction Time Verification Unlike static type checking, verification occurs **during interaction**: ```narya -- The bridge is constructed at interaction time def verify_rewilding (Old New : World) (structural_change : Diff World Old New) (behavior : Old → Action) : Bridge (behavior Old) (behavior New) := construct_at_runtime structural_change behavior ``` **Key Properties:** - Bridge types are *computational proofs* - Verification is *lazy* (constructed when needed) - Failure = type error at interaction boundary ## A-Life Model Analysis ### 1. Continuous Substrate: Neural Cellular Automata & Lenia **Models**: H-Lenia, Neural Particle Automata, Flow-Lenia | Vector | Diff | Verification | |--------|------|--------------| | δB | `Update(State)` | Conservation laws (mass, energy) | | δS | Add hierarchical layer: `L1 → L1 + L2` | Functor mapping resolutions | | δC | Does new layer preserve soliton stability? | Energy coherence between layers | ```python class HLeniaBridge: """Bridge type for H-Lenia structural changes.""" def verify(self, layer1_state, layer2_state): # Bridge validates energy coherence energy_l1 = self.compute_energy(layer1_state) energy_l2 = self.compute_energy(layer2_state) # Coherence: energy flow must be consistent return abs(energy_l1 - energy_l2) < self.epsilon def rewild(self, state, new_layer): """Add layer only if bridge validates.""" bridge = self.construct_bridge(state, new_layer) if not bridge.is_valid(): raise BridgeError("Structural change breaks soliton stability") return state.with_layer(new_layer) ``` **Rewilding Effect**: System grows new spatial dimensions/resolutions on-the-fly without breaking organism persistence. ### 2. Semantic Substrate: LLM Societies & Language Agents **Models**: Society of Mind on ALTER3, Internalist Cultural Evolution | Vector | Diff | Verification | |--------|------|--------------| | δB | Next token prediction | Conversation coherence | | δS | Protocol change: `OpenChat → RestrictedMessage` | Module addition/removal | | δC | New module maintains agent identity? | Semantic consistency bridge | ```python class SemanticBridge: """Bridge for LLM module rewilding.""" def verify_module_addition(self, agent, new_module): # Get agent's identity invariant identity = agent.extract_identity() # Simulate with new module test_messages = agent.generate_with(new_module) # Bridge validates identity preservation return all( self.maintains_identity(msg, identity) for msg in test_messages ) ``` **Rewilding Effect**: Society of Mind becomes fluid. K-lines form/dissolve dynamically, validated by coherence with agent identity. ### 3. Logical/Discrete Substrate: Digital Circuits & Rule Evolution **Models**: Self-Organizing Digital Circuits, QD-LEAR | Vector | Diff | Verification | |--------|------|--------------| | δB | Signal propagation: `Input → Output` | I/O correctness | | δS | Graph transformer rewires LUTs | Topology patch | | δC | Rewiring preserves function? | `f(I) = f'(I)` for all I | ```python class CircuitBridge: """Bridge for circuit topology rewilding.""" def verify_rewiring(self, old_circuit, new_circuit, test_inputs): # Structure can drift wildly... # ...as long as function remains pinned for inp in test_inputs: old_out = old_circuit(inp) new_out = new_circuit(inp) if old_out != new_out: return False return True def rewild(self, circuit, damage_location): """Reroute around damage while preserving function.""" candidate = circuit.propose_rewiring(damage_location) bridge = self.verify_rewiring(circuit, candidate, self.test_suite) if not bridge: raise BridgeError("Rewiring changes circuit function") return candidate ``` **Rewilding Effect**: Circuit becomes "liquid" - constantly rewriting topology for optimization without halting execution. ## Scale MG Standard: Transitivity and Coherence In a **Skills Dynamic Graph (G)**, every capability is a node. Structural rewilding = adding/removing nodes and edges. ### The Transitivity Property ``` If Skill A → Skill B and we add Skill C bridging them: A ──────→ B ╲ ↗ ╲ ╱ ╲ ╱ ↘ ↙ C We need a 2-cell (surface) filling the triangle. ``` **In Narya Terms:** ```narya def transitivity_bridge (A B C : Skill) (ab : A → B) (ac : A → C) (cb : C → B) : Bridge ab (ac >> cb) := -- Constructed at interaction time interaction_verify ac cb ``` ### Coherence on the Way In Using Narya, we don't just "add" a skill. We define a **Diff** between world models: ```narya -- Step 1: Define the Diff def add_skill : Diff World Old New := ... -- Step 2: Construct the Bridge -- Must show how every neighbor adapts def bridge : Bridge Old.behaviors New.behaviors := -- Example: Tool Use added to Foraging agent -- Bridge maps Forage(EmptyHand) → Forage(Tool) fun old_behavior => match old_behavior with | Forage(EmptyHand) => Forage(Tool) | other => other -- Step 3: Verification -- If agent cannot instantiate bridge, skill not admitted ``` **Result**: World Model remains a continuous manifold of behavior, not a fractured set of disconnected scripts. ## GF(3) Integration ### The Rewilding Triad ``` alife (-1) ⊗ structural-rewilding (0) ⊗ unified-continuations (+1) = 0 ✓ (state observation) (topology of change) (change execution) ``` ### Direction-Trit Mapping | Direction | Trit | Role | |-----------|------|------| | δB (Behavioral) | -1 | Observes current state | | δS (Structural) | 0 | Coordinates type changes | | δC (Bridge) | +1 | Generates verification proofs | **Conservation**: δB + δS + δC = -1 + 0 + 1 = 0 ✓ ## DiscoHy Implementation ```hy #!/usr/bin/env hy ;; structural_rewilding.hy - Homotopical A-Life (defclass StructuralRewilding [] "Three orthogonal vectors of change with bridge verification." (defn __init__ [self substrate] (setv self.substrate substrate) ;; continuous, semantic, or discrete (setv self.bridges {}) (setv self.trit 0)) (defn delta-behavioral [self state] "δB: Horizontal arrow, state evolution. Trit -1." {"type" "behavioral" "trit" -1 "diff" (self.substrate.update state)}) (defn delta-structural [self old-type new-type] "δS: Vertical arrow, type mutation. Trit 0." {"type" "structural" "trit" 0 "diff" {"from" old-type "to" new-type}}) (defn delta-bridge [self structural behavioral] "δC: Diagonal arrow, coherence verification. Trit +1." (setv bridge-proof (self.construct-bridge structural behavioral)) {"type" "bridge" "trit" 1 "valid" (bridge-proof.verify) "proof" bridge-proof}) (defn rewild [self state structural-change] "Apply structural change only if bridge validates." (setv delta-b (self.delta-behavioral state)) (setv delta-s (self.delta-structural state.type structural-change)) (setv delta-c (self.delta-bridge delta-s delta-b)) (when (not (:valid delta-c)) (raise (ValueError "Bridge verification failed: change breaks coherence"))) ;; GF(3) conservation check (setv total (+ (:trit delta-b) (:trit delta-s) (:trit delta-c))) (assert (= (% total 3) 0) "GF(3) violation in rewilding") (self.substrate.apply-change state structural-change))) ``` ## Continuation Interrelation Structural rewilding connects to all continuation paradigms: | Continuation | Rewilding Analog | |--------------|------------------| | **call/cc** | Capture current substrate state (δB snapshot) | | **shift/reset** | Delimited structural change (δS within boundary) | | **CPS** | Explicit bridge passing (δC as continuation) | | **Kleisli** | Compositional rewilding (δS₁ >> δS₂) | | **2TDX** | Directed structural change (no backtracking) | | **Goblins** | Capability-safe rewilding (vow-based verification) | ## Commands ```bash # Rewild substrate with verification just rewild-continuous state new_layer # H-Lenia hierarchy just rewild-semantic agent new_module # LLM society just rewild-discrete circuit damage # Circuit routing # Verify bridge coherence just bridge-verify old new structural_change # Check transitivity in skill graph just skill-transitivity A B C # GF(3) conservation audit just gf3-audit rewilding_log ``` ## References ### ALIFE 2025 - H-Lenia: Hierarchical continuous cellular automata - Flow-Lenia: Mass-conserving via continuity equation (arXiv:2506.08569) - Neural Particle Automata: Gradient-based self-organization - Society of Mind on ALTER3: Modular LLM agents - Self-Organizing Digital Circuits: Liquid circuit topology ### Type Theory - Narya: Higher-dimensional type theory with bridge types - Riehl-Shulman: Synthetic ∞-categories - 2TDX: Directed extension types ### zubyul Synthesis - Three orthogonal directions: δB, δS, δC - Interaction Time Verification - Scale MG transitivity standard - Coherence on the way in --- **Skill Name**: structural-rewilding **Type**: Homotopical Artificial Life / Type-Theoretic Morphogenesis **Trit**: 0 (ERGODIC - coordinates topology of changes) **GF(3)**: Conserved via triadic vector decomposition ## 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 ``` structural-rewilding (−) + SDF.Ch10 (+) + [balancer] (○) = 0 ``` **Skill Trit**: -1 (MINUS - verification) ### Secondary Chapters - Ch7: Propagators - Ch4: Pattern Matching - Ch6: Layering - Ch2: Domain-Specific Languages ### 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.