--- name: world-hopping description: Badiou-inspired possible world navigation using triangle inequality constraints, version: 1.0.0 --- # World Hopping: Possible World Navigation World hopping is the art of navigating between **possible worlds** in mathematical/musical/philosophical space. Based on Badiou's ontology and Kripke semantics, with triangle inequality as the fundamental constraint. ## Possible Worlds A **possible world** W is a configuration of: ```ruby class PossibleWorld attr_reader :seed, :epoch, :state, :invariants, :accessibility def initialize(seed:, epoch: 0) @seed = seed # Ontological identity @epoch = epoch # Temporal position @state = compute_state # Current configuration @invariants = [] # What persists across transitions @accessibility = {} # Which worlds are reachable end def compute_state rng = SplitMixTernary.new(@seed + @epoch) { color: SeedMiner.color_at(@seed, @epoch), mathematician: select_mathematician(rng), operation: select_operation(rng), polarity: [:positive, :negative, :neutral][rng.next_ternary + 1] } end end ``` ## Accessibility Relations ### Modal Logic Foundation - **Reflexive**: Every world can reach itself (∀W: W → W) - **Symmetric**: If W₁ → W₂ then W₂ → W₁ (reversible hops) - **Transitive**: If W₁ → W₂ → W₃ then W₁ → W₃ (composable paths) ### Accessibility Matrix ```ruby class AccessibilityRelation def initialize(worlds) @matrix = {} worlds.each do |w1| @matrix[w1.seed] = {} worlds.each do |w2| @matrix[w1.seed][w2.seed] = accessible?(w1, w2) end end end def accessible?(w1, w2) distance = world_distance(w1, w2) max_hop = w1.state[:polarity] == :positive ? 3.0 : 2.0 distance <= max_hop end end ``` ## Badiou's Event Ontology ### Being (L'être) The **void** (∅) underlies all structure. Each world has a void-trace: ```ruby def void_trace(world) # The minimal element that anchors the world world.invariants.min_by(&:complexity) || Void.new end ``` ### Event (L'événement) An **event** is a rupture that creates new possibilities: ```ruby class Event attr_reader :site, :name, :consequences def initialize(site:, name:) @site = site # Where the event occurs @name = name # Self-referential naming @consequences = [] # What follows from the event end def occurs?(world) # Event occurs if site is "on the edge of void" site_elements = world.state.values_at(*@site) site_elements.any? { |e| e.near_void? } end def execute!(world) return unless occurs?(world) # Create new world post-event new_seed = world.seed ^ @name.hash new_world = PossibleWorld.new(seed: new_seed, epoch: world.epoch + 1) # Transfer invariants new_world.invariants = world.invariants.select { |inv| inv.survives?(@name) } new_world end end ``` ### Truth (La vérité) A **truth** is a generic subset that extends from the event: ```ruby class TruthProcedure def initialize(event, world) @event = event @world = world @generic_subset = [] end def extend!(element) # Add element if it's forced by the event if forced?(element) @generic_subset << element propagate_consequences!(element) end end def forced?(element) # Element is forced if it's in every possible extension @world.accessibility.values.all? do |reachable| reachable.state.values.include?(element) || @generic_subset.any? { |g| g.implies?(element) } end end end ``` ## Triangle Inequality Hopping ### Distance Metric ```ruby def world_distance(w1, w2) # Being component: Hamming distance of seeds being = hamming_distance(w1.seed, w2.seed) # Event component: temporal separation event = (w1.epoch - w2.epoch).abs # Truth component: invariant divergence shared = (w1.invariants & w2.invariants).size total = (w1.invariants | w2.invariants).size truth = total > 0 ? 1.0 - (shared.to_f / total) : 0.0 # Weighted Euclidean Math.sqrt(being**2 + event**2 + (truth * 10)**2) end def hamming_distance(a, b) (a ^ b).to_s(2).count('1') end ``` ### Triangle Inequality Constraint ```ruby def valid_hop?(w1, w2, w3) d12 = world_distance(w1, w2) d23 = world_distance(w2, w3) d13 = world_distance(w1, w3) d13 <= d12 + d23 # Triangle inequality end def find_shortest_path(start, target, worlds) # Dijkstra with triangle inequality pruning distances = { start.seed => 0 } previous = {} queue = [start] while queue.any? current = queue.min_by { |w| distances[w.seed] } queue.delete(current) break if current.seed == target.seed worlds.each do |neighbor| next unless accessible?(current, neighbor) d = distances[current.seed] + world_distance(current, neighbor) # Prune if triangle inequality would be violated if distances[neighbor.seed].nil? || d < distances[neighbor.seed] if valid_hop?(start, current, neighbor) distances[neighbor.seed] = d previous[neighbor.seed] = current queue << neighbor end end end end # Reconstruct path path = [] current = target while previous[current.seed] path.unshift(current) current = previous[current.seed] end path.unshift(start) path end ``` ## World Hopping Moves ### 1. SLIDE: Adjacent World Move to a world that differs in one dimension: ```ruby move = WorldHop::Slide.new( from: current_world, dimension: :epoch, direction: :forward ) # Result: epoch += 1, all else preserved ``` ### 2. LEAP: Distant World Jump to a non-adjacent world via event: ```ruby move = WorldHop::Leap.new( from: current_world, event: Event.new(site: [:color], name: "Modulation"), to: target_world ) # Requires: event.occurs?(current_world) ``` ### 3. REFLECT: Dual World Access the contravariant dual: ```ruby move = WorldHop::Reflect.new( from: current_world, duality: :polarity_inversion ) # Result: positive ↔ negative, structure preserved ``` ### 4. COMPOSE: Path Through Intermediate Use triangle inequality for indirect access: ```ruby move = WorldHop::Compose.new( from: w1, via: [w2, w3], # Intermediate worlds to: w4 ) # Requires: d(w1,w4) ≤ d(w1,w2) + d(w2,w3) + d(w3,w4) ``` ## Integration with Music Topos ### Musical World Hopping ```ruby # Worlds are keys/modes c_major = PossibleWorld.new(seed: 0x43, metadata: { key: :C, mode: :major }) a_minor = PossibleWorld.new(seed: 0x41, metadata: { key: :A, mode: :minor }) f_lydian = PossibleWorld.new(seed: 0x46, metadata: { key: :F, mode: :lydian }) # Modulation as event modulation = Event.new( site: [:key, :mode], name: "Pivot chord modulation" ) # Execute hop new_world = modulation.execute!(c_major) ``` ### Mathematician World Hopping ```ruby # Each mathematician inhabits a world ramanujan_world = PossibleWorld.new( seed: 0x1729, metadata: { mathematician: :ramanujan, domain: :number_theory } ) grothendieck_world = PossibleWorld.new( seed: 0x42D, metadata: { mathematician: :grothendieck, domain: :algebraic_geometry } ) # Find path between mathematical worlds path = find_shortest_path(ramanujan_world, grothendieck_world, all_worlds) ``` ### Synadia-Distributed Hopping ```ruby # Publish hop intentions SynadiaBroadcast.publish("world.hop.propose", { from: current_world.seed, to: target_world.seed, event: event.name }) # Consensus on valid hops SynadiaBroadcast.subscribe("world.hop.validate") do |msg| if valid_hop?(current, intermediate, msg.data[:target]) SynadiaBroadcast.publish("world.hop.accept", msg.data) end end ``` ## Philosophical Notes ### Kripke vs Badiou - **Kripke**: Possible worlds are fixed; accessibility is structural - **Badiou**: Events create new possibilities; truth is procedural Our system synthesizes both: - Accessibility matrix (Kripke) provides the topology - Events (Badiou) create new worlds within that topology - Triangle inequality constrains what's reachable ### Connection to Music Musical modulation IS world hopping: - **Key change**: New tonal world with different accessible chords - **Mode change**: Same tonic, different intervallic structure - **Enharmonic reinterpretation**: Same sound, different world ## Integration with Unworld (Derivational vs Temporal) The **unworld** skill replaces temporal succession with derivational chains. World-hopping integrates: ```ruby # Temporal approach: worlds indexed by time temporal_path = [w(t=0), w(t=1), w(t=2)] # Time-ordered sequence # Derivational approach: worlds indexed by seed derivation derivational_path = [ w(seed: 0x42), # Origin w(seed: derive(0x42, :event_A)), # First derivation w(seed: derive(0x42, :event_B)) # Second derivation (parallel!) ] # Key difference: derivational chains can branch without temporal ordering def derive(seed, event) seed ^ event.hash # XOR preserves invertibility end # Unworld mode: check if path is derivationally valid (not just temporally) def valid_derivational_path?(path) path.each_cons(2).all? do |w1, w2| # w2 must be derivable from w1 via some event possible_events.any? { |e| derive(w1.seed, e) == w2.seed } end end ``` ## World Accessibility Graph ```mermaid graph LR subgraph "Temporal Layer" T0((W₀)) --> T1((W₁)) T1 --> T2((W₂)) end subgraph "Derivational Layer" D0((seed:42)) -->|event_A| DA((seed:A7)) D0 -->|event_B| DB((seed:B3)) DA -->|event_C| DAC((seed:E4)) DB -->|event_C| DBC((seed:F1)) end subgraph "Accessibility Relations" W1[("W (C major)")] <-->|"d=1.2"| W2[("W (G major)")] W2 <-->|"d=1.8"| W3[("W (D major)")] W1 -.->|"d≤3.0"| W3 end style D0 fill:#ff6b6b style DA fill:#4ecdc4 style DB fill:#45b7d1 style DAC fill:#96ceb4 style DBC fill:#ffeaa7 ``` ## Triangle Inequality Violation Detection ```ruby class TriangleViolationDetector def initialize(worlds) @worlds = worlds @violations = [] end def detect_all_violations @worlds.combination(3).each do |w1, w2, w3| check_triangle(w1, w2, w3) end @violations end def check_triangle(w1, w2, w3) d12 = world_distance(w1, w2) d23 = world_distance(w2, w3) d13 = world_distance(w1, w3) # Check all three triangle inequalities violations = [] violations << [:d13_exceeds, d12, d23, d13] if d13 > d12 + d23 violations << [:d12_exceeds, d13, d23, d12] if d12 > d13 + d23 violations << [:d23_exceeds, d12, d13, d23] if d23 > d12 + d13 violations.each do |v| @violations << { worlds: [w1.seed, w2.seed, w3.seed], type: v[0], distances: { d12: d12, d23: d23, d13: d13 }, excess: v[3] - (v[1] + v[2]) } end end def report return "✓ No triangle inequality violations" if @violations.empty? @violations.map do |v| "⚠ VIOLATION: #{v[:type]} in worlds #{v[:worlds]}\n" + " Distances: d₁₂=#{v[:distances][:d12]}, d₂₃=#{v[:distances][:d23]}, d₁₃=#{v[:distances][:d13]}\n" + " Excess: #{v[:excess].round(3)}" end.join("\n\n") end end # Example usage: detector = TriangleViolationDetector.new(all_worlds) detector.detect_all_violations puts detector.report # Sample output: # ⚠ VIOLATION: d13_exceeds in worlds [0x42, 0x69, 0xFF] # Distances: d₁₂=2.1, d₂₃=1.8, d₁₃=5.2 # Excess: 1.3 ``` ## Commands ```bash just world-hop from to # Hop between worlds just world-graph # Visualize accessibility graph just world-distance w1 w2 # Calculate world distance just shortest-path w1 w2 # Find optimal hop sequence just event-trigger site name # Create and trigger event just triangle-check # Detect triangle inequality violations just unworld-derive seed event # Derive new world (atemporal) ``` ## 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: 9. Generic Procedures **Concepts**: dispatch, multimethod, predicate dispatch, generic ### GF(3) Balanced Triad ``` world-hopping (+) + SDF.Ch9 (○) + [balancer] (−) = 0 ``` **Skill Trit**: 1 (PLUS - generation) ### Secondary Chapters - Ch3: Variations on an Arithmetic Theme - Ch6: Layering - Ch1: Flexibility through Abstraction - Ch5: Evaluation - Ch10: Adventure Game Example - Ch7: Propagators ### Connection Pattern Generic procedures dispatch on predicates. This skill selects implementations dynamically. ## 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. ## Forward Reference - unified-reafference (goose↔claude↔amp world transitions)