Low-Discrepancy Sequences

Deterministic color generation via low-discrepancy sequences with bijective index recovery.

Purpose

Extends beyond the golden angle (φ) with multiple low-discrepancy sequences for uniform color space coverage. All sequences maintain bijectivity: given a color and seed, you can recover the index n.

Sequences Implemented

1. Golden Angle (φ)

• Dimension: 1D (hue only)
• Uniformity: Optimal for 1D
• Source: φ = (1 + √5)/2
• Formula: hue = (seed + n/φ) mod 1

2. Plastic Constant (φ₂)

• Dimension: 2D (hue + saturation)
• Uniformity: Optimal for 2D
• Source: φ₂ ≈ 1.324717... (root of x³ = x + 1)
• Formula:
  • h = (seed + n/φ₂) mod 1
  • s = (seed + n/φ₂²) mod 1

3. Halton Sequence

• Dimension: nD (direct RGB or HSL)
• Uniformity: Good for any dimension
• Source: Prime bases (2, 3, 5, 7...)
• Formula: halton(n, base) = ∑ dᵢ/baseⁱ⁺¹

4. R-sequence (Recursive)

• Dimension: nD
• Uniformity: Near-optimal
• Source: φ_d (d-dimensional golden ratio)
• Formula: α_d = roots of x^(d+1) = x + 1

5. Kronecker Sequence

• Dimension: 1D
• Uniformity: Optimal (equidistributed)
• Source: Any irrational α
• Formula: {nα} mod 1

6. Sobol Sequence

• Dimension: nD (up to 1000+)
• Uniformity: Excellent for high dimensions
• Source: Direction numbers
• Formula: Gray code XOR with direction vectors

7. Pisot Sequence

• Dimension: nD
• Uniformity: Quasiperiodic
• Source: Pisot-Vijayaraghavan numbers (algebraic integers)
• Formula: θⁿ rounded to nearest integer

8. Continued Fractions

• Dimension: 1D
• Uniformity: Geodesic in hyperbolic geometry
• Source: Continued fraction expansion
• Formula: [a₀; a₁, a₂, ...] convergents

Bijection Property

All sequences are bijective on index: Given (color, seed), you can recover n.

This enables:

• Reafference: "I generated color C at index n"
• Inverse queries: "What index produced this color?"
• Temporal reconstruction: "When did I see this color?"

Integration with Gay.jl

These sequences extend the existing gay-mcp MCP server tools:

• gay_golden_thread: Current φ-based generation
• gay_plastic_thread: New φ₂-based 2D generation
• gay_halton_color: Direct RGB via Halton
• gay_r_sequence: n-dimensional R-sequence
• gay_sobol_color: High-dimensional Sobol
• gay_invert_color: Recover index from color

Related Skills

• gay-mcp: Deterministic color generation foundation
• reafference: Self-recognition via prediction matching
• golden-thread: Original φ spiral implementation
• phenomenal-bisect: Temperature τ bisection using colors
• crystal-family: Crystallographic color assignments
• bidirectional-awareness: Skill graph visualization colors

GF(3) Trit Assignment

Trit: 0 (ERGODIC)

Low-discrepancy sequences are infrastructure for uniform space coverage - neither generative (+1) nor analytical (-1), but foundational coordination (0).

References

1. Niederreiter, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods
2. Kuipers, L. & Niederreiter, H. (1974). Uniform Distribution of Sequences
3. Pisot, C. & Salem, R. (1963). Algebraic Numbers and Fourier Analysis
4. Arnoux, P. & Ito, S. (2001). Pisot substitutions and Rauzy fractals
5. Series, C. (1985). The geometry of Markoff numbers (continued fractions)

Usage Example

using LowDiscrepancySequences

# Golden angle (current method)
color1 = golden_angle_color(69, seed=42)

# Plastic constant (2D: hue + saturation)
color2 = plastic_color(69, seed=42)

# Halton (direct RGB)
color3 = halton_color(69)

# R-sequence (3D)
color4 = r_sequence_color(69, dim=3, seed=42)

# Invert: recover index
n = invert_color(color2, method=:plastic, seed=42)
@assert n == 69

Connections to Geodesics

Continued fractions provide geodesic paths in hyperbolic geometry (PSL(2,ℝ) action on ℍ²). This connects to:

• Geodesic skill representations (shortest execution paths)
• Hyperbolic geometry of skill space
• Non-backtracking paths (prime geodesics)

The Farey sequence F_n = {p/q : gcd(p,q)=1, 0≤p≤q≤n} gives rational approximations to irrationals via continued fractions, mirroring the discrete approximations to geodesic flows.