World Teleportation Abductive Testing

Gay.jl includes an abductive testing framework for exploring and verifying the deterministic color transformations in GayInvaders.

What is Abductive Testing?

Abductive reasoning infers causes from observed effects—the inverse of deduction. Given an observed world color, we reason backwards to determine:

1. Which invader ID produced this color
2. What derangement (RGB channel permutation) was applied
3. What tropical blend parameter t was used

This is analogous to debugging: observing incorrect behavior and reasoning backwards to find the root cause.

Key Concepts

World Teleportation

Each invader "teleports" through a transformation pipeline:

Source Color → Derangement → Tropical Blend → World Color
     ↑              ↑              ↑
   SPI hash    RGB permute    min-plus algebra

All transformations are deterministic (Strong Parallelism Invariance), so the same invader ID + seed always produces the same world color.

Teleportation Properties

The framework tests four key properties:

REPL Commands

The Gay REPL provides interactive world exploration:

!teleport <id>   # Jump to invader's world
!world           # Show current world state  
!back            # Return to previous world
!abduce r g b    # Infer invader from observed RGB
!jump <n>        # Jump to nth hypothesis
!neighbors [r]   # Explore nearby invaders
!test [n]        # Run abductive roundtrip tests

Example Session

gay[0]> !teleport 42
  ⚡ Teleported to world #42
  ─────────────────────────────────────────
  Invader #42
    Source:   ██ (SPI hash)
    Deranged: ██ (perm=1)
    World:    ██ (t=0.42)
    Spin:     ↓ (-1)
  ─────────────────────────────────────────

gay[0]> !neighbors 3
  Neighboring worlds (radius=3):
    [39] ██ → ██ ↑
    [40] ██ → ██ ↓
    [41] ██ → ██ ↑
  → [42] ██ → ██ ↓
    [43] ██ → ██ ↑
    [44] ██ → ██ ↓
    [45] ██ → ██ ↑

gay[0]> !abduce 0.5 0.3 0.7
  🔍 Abducing invader from color...
  Target: ██
  ─────────────────────────────────────────
  Top hypotheses (use !jump <n> to explore):
  [1] ID=12847 conf=0.923 ██
  [2] ID=34291 conf=0.891 ██
  [3] ID=7823 conf=0.887 ██

gay[0]> !jump 1
  🚀 Jumped to hypothesis #1

Programmatic API

Simulate Teleportation

using Gay: simulate_teleportation

# Get full state of an invader's teleportation
sim = simulate_teleportation(42, Gay.GAY_SEED)
# Returns: (id, seed, source, derangement, deranged, world_base, tropical_t, world, spin)

Abductive Inference

using Gay: abduce_invader, abduce_from_source
using Colors: RGB

# Find invaders that could have produced this world color
observed = RGB(0.5, 0.3, 0.7)
hypotheses = abduce_invader(observed; search_range=1:50000, top_k=5)

for h in hypotheses
    println("ID=$(h.id) confidence=$(h.confidence)")
end

# If we have the source color, exact recovery is possible
source = RGB(0.2, 0.4, 0.6)
id = abduce_from_source(source)  # Returns exact ID or nothing

Property Testing

using Gay: test_property, test_all_properties
using Gay: SPIDeterminism, DerangementBijectivity

# Test individual properties
@assert test_property(SPIDeterminism(), 42, Gay.GAY_SEED)
@assert test_property(DerangementBijectivity(), 42, Gay.GAY_SEED)

# Test all properties at once
props = test_all_properties(1337, Gay.GAY_SEED)
# Returns: (spi=true, bijectivity=true, idempotence=true, spin=true)

Roundtrip Tests

using Gay: abductive_roundtrip_test

# Test that we can recover an invader from its world color
@assert abductive_roundtrip_test(42, Gay.GAY_SEED)

# Run many tests
using Gay: run_abductive_tests
results = run_abductive_tests(n_samples=100)

Mathematical Background

Tropical Geometry

The tropical semiring replaces standard arithmetic:

Tropical blending creates hard edges and plateaus, unlike smooth linear interpolation.

Derangements

A derangement is a permutation with no fixed points. For RGB (3 elements), there are exactly 2 derangements:

1. [2, 3, 1]: R→G, G→B, B→R (cyclic left)
2. [3, 1, 2]: R→B, G→R, B→G (cyclic right)

Each invader uses one based on id % 2.

Ising Spin

Each invader has a spin σ ∈ {-1, +1} derived from XOR parity:

spin = ((id ⊻ (id >> 16)) & 1 == 0) ? 1 : -1

This enables magnetization calculations for invader fleets.

Integration with MultipleInterfaces.jl

The TeleportationProperty types are inspired by MultipleInterfaces.jl:

• Properties act like interface predicates
• Multiple properties can be composed and tested
• Dispatch on property types enables extensible testing

abstract type TeleportationProperty end
struct SPIDeterminism <: TeleportationProperty end
struct DerangementBijectivity <: TeleportationProperty end
# ...

# Dispatch on property type
function test_property(::SPIDeterminism, id, seed)
    sim1 = simulate_teleportation(id, seed)
    sim2 = simulate_teleportation(id, seed)
    return sim1.world == sim2.world
end

See Also

• GayInvaders - The invader generation system
• Strong Parallelism Invariance - Core determinism guarantees
• Tropical Geometry - Mathematical background