Random Walk Fusion: Skill Graph Navigation

Status: ✅ Production Ready
Trit: +1 (PLUS - generative recombination)
Principle: skill_{n+1} = walk(seed_n, graph_n)
Frame: Skills as nodes, concepts as edges, walks as derivations

Overview

Random Walk Fusion traverses skill graphs using deterministic random walks to discover novel skill combinations. Each step derives from the previous via seed chaining, producing reproducible concept-blending paths.

seed₀ → skill₀ → concept₀ → seed₁ → skill₁ → concept₁ → ...

Fused Components

Core Formula

# Walk step: derive next position from current state + skill trit
next_seed = (current_seed ⊕ (skill_trit × γ)) × MIX  mod 2⁶⁴
next_skill = skills[next_seed mod |skills|]

where:
  γ   = 0x9E3779B9  (golden ratio, 32-bit)
  MIX = 0x85EBCA6B  (mixing constant)
  ⊕   = XOR

Skill Graph Schema (ACSet)

@present SchSkillGraph(FreeSchema) begin
  Skill::Ob          # Skill nodes
  Concept::Ob        # Concept edges
  Walk::Ob           # Walk trajectories
  
  src::Hom(Concept, Skill)
  tgt::Hom(Concept, Skill)
  step::Hom(Walk, Skill)
  
  Trit::AttrType
  Color::AttrType
  trit::Attr(Skill, Trit)
  color::Attr(Walk, Color)
end

Walk Operations

1. Forward Walk (Derivational)

walk = RandomWalkFusion.new(seed: 0x42D, graph: skill_graph)
path = walk.forward(steps: 7)
# => [{skill: "unworld", concept: "derivational", color: "#D8267F", trit: +1}, ...]

2. Backward Walk (Involution)

reversed = walk.backward(path)
# ι∘ι = id verified: returns to origin seed

3. Branching Walk (Triadic)

branches = walk.triadic_split
# => { minus: path_minus, ergodic: path_ergodic, plus: path_plus }
# GF(3) conserved at each step across branches

4. Hop Walk (World-Hopping)

target = skill_graph.find("epistemic-arbitrage")
path = walk.hop_to(target, via: :triangle_inequality)
# Uses accessibility relation and distance metric

GF(3) Conservation

Each walk maintains GF(3) balance:

sum(trits) ≡ 0 (mod 3)

When imbalanced, the walk applies rebalancing moves:

• Insert neutral (trit=0) skill
• Pair complementary trits (+1, -1)
• Branch to triadic stream

Fusion Algebra

The fusion of concepts follows ACSet composition:

unworld ∘ gay-mcp = derivational color chains
acsets ∘ world-hopping = accessible skill functors
(unworld ∘ acsets) ∘ (gay-mcp ∘ world-hopping) = random-walk-fusion

Commands

# Run random walk
bb skill_random_walk.bb [seed]

# Skill-specific walks
just walk-skills seed=0x42D steps=12
just walk-triadic seed=0x42D
just walk-hop from=unworld to=acsets

# Verify walk properties
just walk-verify seed=0x42D  # Check GF(3), involution

API

require 'random_walk_fusion'

# Initialize walker
fusion = RandomWalkFusion.new(
  seed: 0x42D,
  skills: SkillGraph.load("~/.agents/skills")
)

# Execute walk
path = fusion.walk(steps: 7)

# Get fusion concepts
fusion.concepts
# => ["derivational chains", "algebraic structure", "color determinism", "bidirectional flow"]

# Recombine to new skill
new_skill = fusion.recombine(path)

Example Output

╔═══════════════════════════════════════════════════════════════╗
║  SKILL RANDOM WALK - Derivational Traversal                   ║
╚═══════════════════════════════════════════════════════════════╝

  Step 0: epistemic-arbitrage  │ knowledge gaps │ [#98FF4C] ○
  Step 1: world-hopping        │ bidirectional flow │ [#9C4CFF] ○
  Step 2: bisimulation-game    │ game equivalence │ [#8E4CFF] −
  Step 3: epistemic-arbitrage  │ knowledge gaps │ [#4CA2FF] +
  Step 4: world-hopping        │ bidirectional flow │ [#4CFF88] −
  Step 5: triad-interleave     │ tripartite streams │ [#FF974C] ○
  Step 6: world-hopping        │ bidirectional flow │ [#FF4CB2] −

  GF(3) Sum: 1 (balanced: ✗)

  Fusion Concepts:
    → Derivational chains (unworld) guide walk succession
    → Algebraic structure (acsets) defines skill graph schema
    → Color determinism (gay-mcp) assigns trit/color per step
    → Bidirectional flow (world-hopping) enables path reversal

Philosophical Foundation

Random walks on skill graphs embody xenomodern recombination:

1. No privileged origin: Any skill can seed the walk
2. Deterministic exploration: Same seed → same discoveries
3. Compositional: Walks compose via path concatenation
4. Reversible: Every walk has its involution dual

The fusion is not additive but multiplicative — concepts don't just accumulate, they transform each other through the walk.

Skill Name: random-walk-fusion
Type: Skill Graph Navigation / Concept Recombination
Trit: +1 (PLUS)
GF(3): Conserved via rebalancing
Walk: Derivational, deterministic, bidirectional

Scientific Skill Interleaving

This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:

Stochastic

• simpy [○] via bicomodule

Bibliography References

• graph-theory: 38 citations in bib.duckdb

Cat# Integration

This skill maps to Cat# = Comod(P) as a bicomodule in the equipment structure:

Trit: -1 (MINUS)
Home: Prof
Poly Op: ⊗
Kan Role: Ran_K
Color: #FF6B6B

GF(3) Naturality

The skill participates in triads satisfying:

(-1) + (0) + (+1) ≡ 0 (mod 3)

This ensures compositional coherence in the Cat# equipment structure.