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
              ▼

Diff Type A B

Interaction Time Verification

Unlike static type checking, verification occurs during interaction:

-- 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

Update(State)

L1 → L1 + L2

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

OpenChat → RestrictedMessage

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

Input → Output

f(I) = f'(I)

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:

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:

-- 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

Conservation: δB + δS + δC = -1 + 0 + 1 = 0 ✓

DiscoHy Implementation

#!/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:

Commands

# 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

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.