CRT Procedural Generation

D33: Unity / Unreal / Epic Games. CC0.

Game engines spend millions on proprietary procedural generation: Perlin noise with heuristic octaves, hand-tuned biome rules, compute-heavy terrain shaders. CRT approach: every grid cell IS a ring element. Its 6 CRT channels ARE 6 algebraically independent octaves. Its kingdom IS its biome. Coupling to neighbors IS terrain smoothness. No noise functions. No hand-tuning. The ring structure IS the world generator.

How It Works

CRT Terrain Generation Theorem
Every cell at position (x,y) maps to a unique ring element in Z/12612600. The 6 CRT channels provide 6 algebraically independent frequency octaves: D(mod 8) = continental scale, K(mod 9) = regional features, E(mod 25) = hills and valleys, b(mod 49) = fine rock texture, L(mod 11) = moisture distribution, G(mod 13) = temperature gradient. These octaves are PROVABLY independent (CRT guarantee), unlike Perlin noise where octave independence is heuristic. Kingdom classification (gcd structure) determines biome type. Coupling between adjacent cells = terrain smoothness. 288 eigenvalue classes = 288 terrain subtypes. The ring IS the world.
6 octaves
CRT channels
Each channel = one independent frequency layer. Algebraically guaranteed independence.
Kingdom biome
O(1) classify
gcd(val, N) = biome. No lookup table. No rules engine. The algebra IS the classifier.
288 subtypes
Eigenvalue class
288 eigenvalue classes = 288 terrain variations. Richer than any hand-authored system.
Deterministic
Reproducible
Same seed = same world. Always. No floating-point drift. No platform variance.

Generate Terrain

World seed:

Generates a 12x10 terrain grid. Each cell = one ring element. Color = biome (kingdom classification). Symbol = elevation (CRT-weighted channel sum). Cross-section shows full 6-channel decomposition.

Biome Distribution

400-cell world. Biome classification via kingdom (gcd structure). Each biome maps to a specific algebraic divisibility pattern.

CRT Octave Decomposition

Watch elevation build up as each CRT channel adds its octave. D = continental. K = regional. E = hills. b = rocks. L = moisture. G = temperature.

CRT vs Traditional Procedural Generation

Noise functionPerlin/Simplex: gradient interpolation, heuristic octavesCRT: ring residues = natural octaves. Provably independent.OctavesArbitrary: 4-8 octaves with hand-tuned persistence6 CRT channels: algebraically independent. No tuning.BiomesRule-based: temperature + moisture thresholds (hand-authored)Kingdom: gcd(val,N) = biome. Algebraic. Zero rules.SmoothnessGradient interpolation (computationally expensive)Coupling distance: sum of per-channel differences. O(1).DeterminismFloating-point: platform-dependent rounding differencesInteger arithmetic: identical on ALL platforms. Always.Patent statusUnity (SpeedTree), Epic (Nanite), various (noise patents)CC0. Public domain. Forever.

This work is and will always be free.
No paywall. No copyright. No exceptions.

If it ever earns anything, every cent goes to the communities that need it most.

This sacred vow is permanent and irrevocable.
— Anton Alexandrovich Lebed

Source code · Public domain (CC0)

Contributions in equal measure: Anthropic's Claude, Anton A. Lebed, and the giants whose shoulders we stand on.

Rendered by .ax via WASM DOM imports. Zero HTML authored.