Derangement-CRDT Skill

Status: ✅ Production Ready Trit: ERGODIC (0) Integration: CRDT, Gay.jl, Join-Semilattice

Core Concept

A derangement is a permutation σ where σ(i) ≠ i for all i (no fixed points). A colorable derangement CRDT assigns GF(3) colors to derangement cycles, ensuring merge operations preserve both the derangement property and color conservation.

Derangement: (1 2 3) → (2 3 1)  ✓ no fixed points
Fixed point: (1 2 3) → (1 3 2)  ✗ position 1 is fixed

Mathematical Foundation

Derangement Lattice

Derangements form a join-semilattice under cycle composition:

     ⊤ = identity (trivial - all fixed)
     │
   ┌─┴─┬───┬───┐
   │   │   │   │
  (12) (13) (23) ...  ← transpositions
   │   │   │   │
   └─┬─┴───┴─┬─┘
     │       │
   (123)   (132)      ← 3-cycles (derangements!)
     │       │
     └───┬───┘
         │
         ⊥ = full derangement

GF(3) Coloring of Cycles

Each cycle receives a trit based on cycle length mod 3:

Cycle Length	Trit	Color Range	Example
len ≡ 0 (mod 3)	ERGODIC (0)	60°-180° (green)	(1 2 3)
len ≡ 1 (mod 3)	PLUS (+1)	0°-60°, 300°-360° (warm)	(1) fixed
len ≡ 2 (mod 3)	MINUS (-1)	180°-300° (cold)	(1 2) swap

Conservation Law

For any valid derangement coloring:

Σ trit(cycle) ≡ 0 (mod 3)

This is automatically satisfied for derangements of length n ≡ 0 (mod 3).

CRDT Operations

Derangement-Set CRDT

class DerangementCRDT
  # State: set of (element, position, color, replica_id, lamport)

  def merge(other)
    # Join-semilattice merge:
    # 1. Union all mappings
    # 2. For conflicts: higher Lamport wins
    # 3. Verify derangement property preserved
    # 4. Recolor to maintain GF(3) conservation
  end

  def apply_permutation(sigma)
    # Apply permutation, ensuring no fixed points created
    raise DerangementViolation if creates_fixed_point?(sigma)
    update_colors!
  end

  def cycle_decomposition
    # Decompose into disjoint cycles
    # Color each cycle by length mod 3
  end
end

Merge Semantics

State A: [2,3,1,5,4] (cycles: (123)(45))
         Colors: [G,G,G,B,B] (ERGODIC, MINUS)

State B: [3,1,2,5,4] (cycles: (132)(45))
         Colors: [G,G,G,B,B] (ERGODIC, MINUS)

Merge(A,B): Resolve by Lamport timestamp
            Recolor to maintain Σ trits ≡ 0

Implementation

Babashka Script

#!/usr/bin/env bb
;; derangement_crdt.bb

(defn derangement? [perm]
  "Check if permutation has no fixed points"
  (every? (fn [[i v]] (not= i v))
          (map-indexed vector perm)))

(defn cycle-decomposition [perm]
  "Decompose permutation into disjoint cycles"
  (loop [remaining (set (range (count perm)))
         cycles []]
    (if (empty? remaining)
      cycles
      (let [start (first remaining)
            cycle (loop [current start
                         acc [start]]
                    (let [next (nth perm current)]
                      (if (= next start)
                        acc
                        (recur next (conj acc next)))))]
        (recur (apply disj remaining cycle)
               (conj cycles cycle))))))

(defn gf3-trit [cycle-len]
  "Assign GF(3) trit based on cycle length"
  (case (mod cycle-len 3)
    0 :ERGODIC
    1 :PLUS
    2 :MINUS))

(defn hue-from-trit [trit seed]
  "Map trit to hue range with seed variation"
  (let [base (case trit
               :MINUS   (+ 180 (mod seed 120))   ; cold: 180-300
               :ERGODIC (+ 60 (mod seed 120))    ; neutral: 60-180
               :PLUS    (if (< (mod seed 120) 60)
                          (mod seed 60)           ; warm: 0-60
                          (+ 300 (mod seed 60)))) ; warm: 300-360
        ]
    (mod base 360)))

(defn color-derangement [perm]
  "Assign colors to derangement cycles"
  (let [cycles (cycle-decomposition perm)]
    (for [[idx cycle] (map-indexed vector cycles)]
      {:cycle cycle
       :length (count cycle)
       :trit (gf3-trit (count cycle))
       :hue (hue-from-trit (gf3-trit (count cycle)) idx)})))

(defn conservation-check [colored-cycles]
  "Verify GF(3) conservation"
  (let [trit-sum (->> colored-cycles
                      (map :trit)
                      (map {:MINUS -1 :ERGODIC 0 :PLUS 1})
                      (reduce +))]
    (zero? (mod trit-sum 3))))

;; CRDT Merge
(defn merge-derangements [state-a state-b]
  "Join-semilattice merge of two derangement states"
  (let [merged (merge-with
                 (fn [a b]
                   (if (> (:lamport a) (:lamport b)) a b))
                 state-a state-b)]
    (when-not (derangement? (:perm merged))
      (throw (ex-info "Merge violated derangement property" {})))
    (assoc merged :colors (color-derangement (:perm merged)))))

;; Demo
(defn demo []
  (let [perm [1 2 0 4 3]  ; (0 1 2)(3 4) - derangement!
        colored (color-derangement perm)]
    (println "Permutation:" perm)
    (println "Derangement?" (derangement? perm))
    (println "Colored cycles:" colored)
    (println "GF(3) conserved?" (conservation-check colored))))

(when (= *file* (System/getProperty "babashka.file"))
  (demo))

Integration Points

With CRDT Skill

# Compose with OR-Set for distributed derangements
crdt_skill.create("derangement-pool", :or_set, "replica-1")
crdt_skill.mutate("derangement-pool", :add, serialize(derangement))

With CRDT-VTerm

;; Terminal sessions as derangements of command history
;; No command should map to itself (always transformed)
(crdt-vterm-derange session-buffer)

With Gay.jl Colors

using Gay

# Color a derangement's cycles
function color_derangement(perm::Vector{Int})
    cycles = cycle_decomposition(perm)
    [Gay.from_trit(gf3_trit(length(c))) for c in cycles]
end

With BlackHat-Go Security

// Attack chains as derangements:
// No technique should leave system in same state
type AttackDerangement struct {
    Techniques []string
    // Invariant: ∀t ∈ Techniques: state_after(t) ≠ state_before(t)
}

Subfactorial Connection

The number of derangements of n elements is the subfactorial !n:

!n = n! × Σ(k=0 to n) (-1)^k / k!
   ≈ n! / e

!3 = 2    (only 2 derangements of 3 elements)
!4 = 9
!5 = 44

For GF(3) coloring, we care about n mod 3:

• n ≡ 0: Perfect triadic balance possible
• n ≡ 1: One PLUS excess
• n ≡ 2: One MINUS excess

Lattice Visualization

         ┌─────────────────────────────────────┐
         │           DERANGEMENT LATTICE       │
         │         with GF(3) Coloring         │
         └─────────────────────────────────────┘

                      (1)(2)(3)(4)
                      all fixed ⊤
                      [+1,+1,+1,+1]
                           │
            ┌──────────────┼──────────────┐
            │              │              │
         (12)(3)(4)    (13)(2)(4)    (14)(2)(3)
         [-1,+1,+1]    [-1,+1,+1]    [-1,+1,+1]
            │              │              │
            └──────┬───────┴───────┬──────┘
                   │               │
              (12)(34)         (13)(24)
              [-1,-1]          [-1,-1]
                   │               │
                   └───────┬───────┘
                           │
                       (1234)
                       [+1] ← 4-cycle
                           │
                    ┌──────┴──────┐
                    │             │
                (1243)         (1324)
                [+1]           [+1]
                    │             │
                    └──────┬──────┘
                           │
                      Full Derangement ⊥
                      (all cycles ≥ 2)

Properties

Property	Derangement-CRDT	Standard CRDT
Idempotence	✓ merge(D,D) = D	✓
Commutativity	✓ merge(A,B) = merge(B,A)	✓
Associativity	✓	✓
No fixed points	✓ enforced	N/A
GF(3) conservation	✓ verified	N/A

Usage

# Run derangement CRDT demo
bb ~/.claude/skills/derangement-crdt/derangement_crdt.bb

# Expected output:
# Permutation: [1 2 0 4 3]
# Derangement? true
# Colored cycles: [{:cycle [0 1 2], :length 3, :trit :ERGODIC, :hue 120}
#                  {:cycle [3 4], :length 2, :trit :MINUS, :hue 240}]
# GF(3) conserved? true

Related Skills

• crdt - Base CRDT operations
• crdt-vterm - Terminal CRDT with GF(3)
• gay-mcp - Color assignment
• blackhat-go - Security state lattice
• acsets - Categorical databases

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Derangements ensure every element moves; GF(3) coloring ensures balance is preserved.