--- name: epistemic-arbitrage description: Propagator-based parallel structure for exploiting knowledge differentials version: 1.0.0 --- # Epistemic Arbitrage: Propagator-Based Knowledge Synthesis Epistemic arbitrage exploits **knowledge differentials** between domains. When concept A in domain X is well-understood, but its isomorphic counterpart A' in domain Y is mysterious, we can **arbitrage** by transferring understanding. ## Core Architecture ### Propagator Networks (Radul/Sussman) A propagator network consists of: ``` ┌─────────┐ ┌─────────────┐ ┌─────────┐ │ Cell A │────▶│ Propagator │────▶│ Cell B │ │ (known) │ │ (transfer) │ │(derived)│ └─────────┘ └─────────────┘ └─────────┘ ``` - **Cells**: Containers for partial information - **Propagators**: Functions that combine/transform information - **Merging**: Monotonic accumulation (information only increases) ### SplitMixTernary RNG For parallel propagation with determinism: ```ruby class SplitMixTernary GOLDEN = 0x9e3779b97f4a7c15 # Golden ratio constant def initialize(seed) @state = seed end def next_ternary # Advance state (SplitMix64) @state = (@state + GOLDEN) & 0xFFFFFFFFFFFFFFFF z = @state z = ((z ^ (z >> 30)) * 0xbf58476d1ce4e5b9) & 0xFFFFFFFFFFFFFFFF z = ((z ^ (z >> 27)) * 0x94d049bb133111eb) & 0xFFFFFFFFFFFFFFFF z = z ^ (z >> 31) # Map to ternary: -1, 0, +1 (z % 3) - 1 end def split(index) # Create independent child stream child_seed = @state ^ (index * GOLDEN) SplitMixTernary.new(child_seed) end end ``` ## Arbitrage Patterns ### Pattern 1: Domain Transfer ```ruby # Knowledge in domain A cell_a = Cell.new(:music, :circle_of_fifths) cell_a.add_info(:structure, :cyclic_group_12) cell_a.add_info(:generator, :perfect_fifth) # Unknown in domain B cell_b = Cell.new(:number_theory, :modular_arithmetic) # Propagator transfers structure propagator = Propagator.new(:domain_transfer) do |source, target| if source.has?(:structure) target.merge(:structure, source.get(:structure)) target.merge(:insight, "Z/12Z isomorphic to circle of fifths") end end propagator.connect(cell_a, cell_b) propagator.fire! ``` ### Pattern 2: Dual Discovery ```ruby # Known: Fourier transform on functions cell_time = Cell.new(:analysis, :time_domain) cell_freq = Cell.new(:analysis, :frequency_domain) # Propagator: Fourier duality duality_propagator = Propagator.new(:fourier_duality) do |time, freq| time.each_property do |prop, val| dual_prop = fourier_dual(prop) # convolution ↔ multiplication, etc. freq.merge(dual_prop, val) end end ``` ### Pattern 3: Triangle Arbitrage When three domains have pairwise connections, exploit the triangle: ```ruby # Three domains with partial knowledge cell_math = Cell.new(:mathematics, :group_theory) cell_music = Cell.new(:music, :harmony) cell_physics = Cell.new(:physics, :symmetry) # Pairwise propagators math_music = Propagator.new(:pitch_class_groups) music_physics = Propagator.new(:overtone_series) physics_math = Propagator.new(:lie_groups) # Triangle inequality enables arbitrage: # d(math, physics) ≤ d(math, music) + d(music, physics) # # If direct math→physics is hard, go via music! ``` ## Local Scoped Propagators ### Scoping for Parallelism Each propagator runs in a local scope with: ```ruby class ScopedPropagator def initialize(name, rng_seed, &block) @name = name @rng = SplitMixTernary.new(rng_seed) @block = block @local_cells = {} end def fork(n) # Create n parallel scopes with independent RNG streams n.times.map do |i| child_rng = @rng.split(i) ScopedPropagator.new("#{@name}/#{i}", child_rng.state, &@block) end end def run(inputs) # Execute in isolated scope @local_cells = inputs.dup @block.call(self, @local_cells) @local_cells end end ``` ### Parallel Arbitrage Network ```ruby # Master network network = ArbitrageNetwork.new(seed: 1069) # Add cells for each domain network.add_cell(:category_theory, knowledge: 0.9) network.add_cell(:music_theory, knowledge: 0.7) network.add_cell(:physics, knowledge: 0.5) network.add_cell(:philosophy, knowledge: 0.3) # Add arbitrage propagators network.add_propagator(:category_to_music, [:category_theory], [:music_theory]) network.add_propagator(:music_to_physics, [:music_theory], [:physics]) network.add_propagator(:physics_to_philosophy, [:physics], [:philosophy]) # Run in parallel (SPI-compliant) results = network.run_parallel(n_workers: 4) ``` ## Integration with Music Topos ### With World Broadcast ```ruby # Each mathematician is a knowledge source broadcasters = WorldBroadcast::TripartiteSystem.new([:ramanujan, :grothendieck, :euler]) # Create cells from mathematician expertise cells = broadcasters.agents.map do |agent| Cell.new(agent.profile[:domain], knowledge: agent.history.size, operations: agent.profile[:operations]) end # Arbitrage between mathematicians network = ArbitrageNetwork.from_cells(cells) network.add_all_pairwise_propagators network.run! ``` ### With Synadia ```ruby # Publish arbitrage opportunities network.on_arbitrage do |source, target, gain| SynadiaBroadcast.publish("arbitrage.opportunity", { from: source.domain, to: target.domain, knowledge_gain: gain }) end # Subscribe to external knowledge SynadiaBroadcast.subscribe("knowledge.*") do |msg| domain = msg.subject.split('.').last.to_sym network.cells[domain].merge(:external, msg.data) end ``` ### With Glass Bead Game ```ruby # Arbitrage opportunities become game moves network.on_arbitrage do |source, target, gain| if gain > threshold move = GlassBeadGame::Connect.new( from: Bead.new(source.domain, source.best_concept), to: Bead.new(target.domain, target.mystery), via: :epistemic_transfer, value: gain ) game.propose_move(move) end end ``` ## Guarantees 1. **SPI Compliance**: Parallel execution ≡ sequential (bitwise identical) 2. **Determinism**: Same seed → same arbitrage discoveries 3. **Monotonicity**: Knowledge only increases (no forgetting) 4. **Locality**: Propagators only access explicitly connected cells 5. **Triangle Inequality**: d(A,C) ≤ d(A,B) + d(B,C) for all knowledge transfers ## Commands ```bash just arbitrage # Run arbitrage network just arbitrage-domains a b # Arbitrage between specific domains just propagator-network # Visualize propagator network just knowledge-flow # Show knowledge flow diagram ``` ## 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 ``` epistemic-arbitrage (+) + SDF.Ch10 (+) + [balancer] (+) = 0 ``` **Skill Trit**: 1 (PLUS - generation) ### Secondary Chapters - Ch7: Propagators - Ch3: Variations on an Arithmetic Theme - Ch4: Pattern Matching - Ch6: Layering ### 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.