--- name: derangement-reflow description: "derangement-reflow skill" version: 1.0.0 --- # Derangement Reflow: World Operators as Information Reflow ## Core Insight **World operators are information reflow operators** because the derangement constraint σ(i)≠i prevents information stasis. Every bit must flow to a *different* position—no self-loops in the information graph. ## The GitHub Blind Spot PR reviews exhibit a **fixed-point pathology**: validators often self-validate their own patterns. This violates the fundamental derangement constraint that enables healthy information flow: ``` ❌ FIXED POINT (σ(i)=i): Validator A → validates → Validator A's output ✓ DERANGEMENT (σ(i)≠i): Validator A → validates → Generator B's output Generator B → generates → for Coordinator C Coordinator C → routes → to Validator A ``` ## GF(3) Reflow Accounting ``` MINUS (−1): Information leaves position (entropy source) - VALIDATORS ERGODIC (0): Information transits (channel) - COORDINATORS PLUS (+1): Information arrives (entropy sink) - GENERATORS Conservation: Σ trits ≡ 0 (mod 3) under all world operators ``` ## Tropical Geometry of Interaction Skill composition paths analyzed via **min-plus semiring** (R ∪ {∞}, min, +): ```python # Tropical distance between skills def tropical_distance(path: list[Skill]) -> float: """ In tropical geometry, shortest path = minimum sum. Path cost = Σ |trit_i - trit_{i+1}| Optimal interleaving minimizes tropical distance while maintaining derangement (no consecutive same-trit skills). """ if len(path) < 2: return 0 cost = 0 for i in range(len(path) - 1): # Derangement check: consecutive skills must differ if path[i].trit == path[i+1].trit: return float('inf') # Invalid path (fixed point) cost += abs(path[i].trit - path[i+1].trit) return cost ``` ## Joint World Modeling via Active Inference The missing nuance in GitHub workflows: **agents must share a joint world model**, not just pass artifacts: ``` ┌─────────────────────────────────────────────────────────────────┐ │ JOINT WORLD MODEL (Active Inference) │ ├─────────────────────────────────────────────────────────────────┤ │ │ │ Each agent maintains: │ │ - μ: belief about world state │ │ - π: policy (preferred future states) │ │ - F: free energy = surprise + complexity │ │ │ │ Derangement ensures: │ │ - Agent A's μ influences Agent B's π (not A's own π) │ │ - Reflow minimizes joint free energy, not individual F │ │ │ │ WEV = ∫ min(F_joint) over world transitions │ │ = value extracted from mandatory redistribution │ │ │ └─────────────────────────────────────────────────────────────────┘ ``` ## Task Interleaving Pattern From thread analysis, the optimal pattern for GF(3) triadic task interleaving: ``` Step 0: Spawn MINUS/ERGODIC/PLUS sub-agents in parallel Step 1: Each agent works on disjoint task subset Step 2: ERGODIC collects results from MINUS and PLUS Step 3: Derangement shuffle: MINUS validates PLUS output PLUS generates from MINUS constraints ERGODIC routes neither to itself Step 4: Merge with GF(3) conservation check ``` ## Implementation ```python #!/usr/bin/env python3 """Derangement reflow validator for skill triplets.""" from dataclasses import dataclass from typing import List, Tuple import json @dataclass class Skill: name: str trit: int # -1, 0, +1 def is_derangement(permutation: List[int]) -> bool: """Check if permutation has no fixed points.""" return all(p != i for i, p in enumerate(permutation)) def validate_triplet_reflow(minus: Skill, ergodic: Skill, plus: Skill) -> dict: """ Validate that triplet follows derangement reflow: - MINUS (-1) must flow to PLUS (+1), not to itself - PLUS (+1) must flow to MINUS (-1), not to itself - ERGODIC (0) routes between, never self-loops """ # GF(3) conservation check trit_sum = minus.trit + ergodic.trit + plus.trit conserved = (trit_sum % 3) == 0 # Derangement check: information flows must cross trit boundaries # MINUS validates → PLUS output (not MINUS) # PLUS generates → from MINUS constraints (not PLUS) # ERGODIC routes → between others (not self) reflow_valid = ( minus.trit != plus.trit and # Different sources minus.trit != ergodic.trit and plus.trit != ergodic.trit ) # Tropical distance for this triplet path = [minus, ergodic, plus] tropical_cost = sum(abs(path[i].trit - path[i+1].trit) for i in range(2)) return { "conserved": conserved, "derangement_valid": reflow_valid, "tropical_cost": tropical_cost, "wev_extractable": conserved and reflow_valid, "triplet": [minus.name, ergodic.name, plus.name], "trits": [minus.trit, ergodic.trit, plus.trit] } def main(): # Example: validate the token-rent-validator triplet from PR #33 accept_no_substitutes = Skill("accept-no-substitutes", -1) skill_creator = Skill("skill-creator", 0) tree_sitter = Skill("tree-sitter", +1) result = validate_triplet_reflow( accept_no_substitutes, skill_creator, tree_sitter ) print(json.dumps(result, indent=2)) # Check if PR #33 has derangement-aware validation code_review = Skill("code-review", -1) narya_proofs = Skill("narya-proofs", -1) # This is a VIOLATION: two validators (-1, -1) = fixed point potential violation_check = { "issue": "Two validators can self-validate each other's patterns", "fix": "Interleave with generator or coordinator between validators", "tropical_cost": float('inf') if code_review.trit == narya_proofs.trit else 0 } print("\n⚠️ PR #33 Derangement Issue:") print(json.dumps(violation_check, indent=2)) if __name__ == "__main__": main() ``` ## GF(3) Triads with This Skill ``` derangement-reflow (0) ⊗ accept-no-substitutes (-1) ⊗ chromatic-walk (+1) = 0 ✓ derangement-reflow (0) ⊗ active-inference (-1) ⊗ world-hopping (+1) = 0 ✓ derangement-reflow (0) ⊗ bisimulation-game (-1) ⊗ glass-bead-game (+1) = 0 ✓ ``` ## References - Derangements on 3 elements ≅ Z/3Z (cyclic group) - Tropical geometry: min-plus semiring for path optimization - Active inference: Friston's free energy principle for joint world modeling - AtomicDerangement3 in signal-mcp/src/loom_failures.rs - perception_matrix_alpha.swift: Sattolo's algorithm for guaranteed derangement