Theoretical Foundations

Strong Parallelism Invariance (SPI)

Gay.jl is built on the Strong Parallelism Invariance pattern from Pigeons.jl, which guarantees reproducible results regardless of parallelization strategy.

The Problem

Standard random number generators (RNGs) produce different results when:

• Threads execute in different orders
• Number of workers changes
• Sequential vs parallel execution

Standard RNG (non-deterministic):
  Thread 1: rand() → 0.42
  Thread 2: rand() → 0.73
  Total depends on which thread calls first!

The Solution: Splittable Streams

Splittable random streams solve this by creating independent sub-streams:

Splittable RNG (deterministic):
  Parent RNG (seed=1069)
      ↓
  split() → Child 1 (always same stream)
      ↓
  split() → Child 2 (always same stream)
      ↓
  split() → Child 3 (always same stream)

Each split() operation:

1. Deterministically derives a child stream from the parent
2. Mutates the parent to advance its state
3. Guarantees the same child for the same split sequence

Mathematical Foundation

The splitting operation uses cryptographic hashing to derive child states:

child_state = hash(parent_state || split_counter)

This ensures:

• Determinism: Same parent state → same child state
• Independence: Child streams are statistically independent
• Irreversibility: Cannot recover parent from child

Color Theory

Wide-Gamut Color Spaces

Gay.jl supports three color spaces with increasing gamut size:

              ┌─────────────────────────────────────┐
              │         CIE 1931 xy Diagram         │
              │                                     │
              │            ╱╲                       │
              │           ╱  ╲  Visible Gamut       │
              │          ╱    ╲                     │
              │         ╱  R   ╲ ← Rec.2020         │
              │        ╱   ╱╲   ╲                   │
              │       ╱   ╱  ╲   ╲ ← Display P3    │
              │      ╱   ╱ s  ╲   ╲                 │
              │     ╱   ╱ RGB  ╲   ╲                │
              │    ╱___╱________╲___╲               │
              │   G              B                  │
              └─────────────────────────────────────┘

Golden Angle Palette Generation

next_palette() uses the golden angle (≈137.5°) for optimal color separation:

φ = (1 + √5) / 2 ≈ 1.618  (golden ratio)
θ = 360° / φ² ≈ 137.5°    (golden angle)

Colors are placed at intervals of θ around the HSV hue wheel, maximizing perceptual distance.

Comrade.jl Sky Models

VLBI Imaging

Very Long Baseline Interferometry (VLBI) reconstructs images from sparse Fourier measurements. The Event Horizon Telescope uses this to image black holes.

Gay.jl's sky models are simplified versions of Comrade.jl's primitives:

Model Composition

Sky models form a semiring under addition and multiplication:

(M, +, *, 0, 1) where:
  + = additive blending (intensity sum)
  * = multiplicative masking (intensity product)
  0 = zero intensity everywhere
  1 = unit intensity everywhere

This allows compositional model building:

m87 = ring + gaussian           # Additive
masked = ring * crescent        # Multiplicative
complex = (ring1 + ring2) * disk  # Combined

Aesthetic Scoring

Models are scored using:

score = 0.3 × coverage + 0.2 × contrast + 0.3 × symmetry + 0.2 × golden_bonus

where:
  coverage = fraction of non-zero pixels
  contrast = std(intensities)
  symmetry = 1 - std(radii) / mean(radii)
  golden_bonus = 1 / |ring_count / φ - 1|

The golden bonus rewards ring counts near the golden ratio (≈1.618).