world-hopping

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

```

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

```

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

```

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(being2 + event2 + (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

```

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)

```

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

```

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

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

```

```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

```

```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

```

```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)

```

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

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.

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.

• unified-reafference (goose↔claude↔amp world transitions)