UUON Recursive Fractal Engine

UUON Foundation Inc. — Phillip Aguilar Ruiz III License: USAL-1.0 · Version: 1.0.0

A GPU-accelerated complex dynamical systems renderer built on WebGL2. Live at: https://uuon-foundation.github.io/recursive-fractal-engine/

What It Is

A parameterized boundary mapper. Every frame is the stability boundary of a complex dynamical system — the exact edge where iteration neither collapses to a fixed point nor escapes to infinity. The engine exposes that boundary as a composable pipeline of interchangeable mathematical operations.

22 pre-transforms × 16 generators × 11 post-transforms × 2 phase-space modes = 7,744 distinct dynamical systems from one compiled shader.

Biological Analog

The retina of a nervous system that sees mathematical structure.

Where other UUON engines model geometry (WFE), state propagation (Propagation Engine), or binary boundary systems (BSE), the Recursive Fractal Engine maps the boundary between order and chaos across parameterized complex dynamical systems. That is what a sensory organ does: it sits at the threshold of signal and noise and reports on the boundary.

Integrated into uuon-clouud, this engine becomes the visual cortex of the system — not a display, but a generator of structured perceptual fields grounded in the mathematics of natural form.

Architecture

Transform Stack

[Pre-transform] → [Symmetry fold] → [Generator loop] → [Post-transform] → [Coloring]

Each layer is independently swappable. The unified contract: every operation consumes and produces a complex number z ∈ ℂ with a fixed parameter set { c, power, pa, pb, symk, escape }.

Pre-transforms (22)

One-shot coordinate warps applied before iteration, or per-loop fold/warp operations applied inside the iteration. Includes: topology inversion (1/z), Möbius transforms, burning ship fold (|Re|+i|Im|), log/exp warps, sin-based warps, polar radial and symmetry multiplications.

Generators (16)

The core iteration rule z → f(z,c). Polynomial family (z²+c through zⁿ+c), transcendental family (eᶻ, sin, cos, tan, sinh, cosh, log), iterative root methods (Newton, Nova), Biomorph, and two original quantum generators.

Post-transforms (11)

Applied once to the final z before coloring. Cosmetic remappings: fractional power, complex power, Möbius post-warp, rational remap, sine/tangent warps, log-spiral, mirror inversion, Kleinian Möbius.

Coloring (20 modes)

Smooth escape, banded, orbit traps (circle/cross/point), final angle/magnitude, stripe average, distance estimate, Lyapunov exponent, geometry normal, tangent/surface flow, metallic, glass/chromatic IOR, negative filter, iridescent/thin film, depth fog, normal map 3D lit, and two original quantum coloring modes.

Original IP

See NOTICE for the full provenance record. Summary of original contributions:

| Component | Description | |-----------|-------------| | Composable transform stack | 22×16×11×2 dynamical systems from one shader via unified (z,c) contract | | K-fold symmetry + rotation | Continuous sym_rot sweeps symmetry axes — not locked to cardinal positions | | safe_pre() guard system | Epsilon-guarded singularity handling for all 22 pre-transform cases | | Quantum Wave generator (14) | ψ → ψ·e^(i|ψ|²)+c — Born-rule probability density drives rotation | | Schrödinger Oscillator (15) | Self-normalizing amplitude oscillator as an iterative feedback rule | | Probability Density coloring (18) | Orbit-accumulated ∫|ψ|²dt mapped to color — quantum statistics applied to fractal rendering | | Phase Portrait coloring (19) | Probability-weighted mean phase mapped to hue — angular momentum channels | | Parameter provenance protocol | Every attractor state is uniquely recoverable from its JSON reproduction key |

Individual generator equations (Mandelbrot, Julia, Newton, Burning Ship, Biomorph, etc.) are prior art. The architecture is not.

Parameter Seed

Every attractor is uniquely identified by its parameter record. Example:

{
  "mode": 1,
  "generator": 14,
  "pretransform": 0,
  "posttransform": 0,
  "iter": 112,
  "escape": 4.0,
  "power": 2.0,
  "symk": 5,
  "sym_rot": 0,
  "tessellation": 0,
  "cx": 0.0,
  "cy": 0.0,
  "zoom": 0.0,
  "julia_re": -0.4,
  "julia_im": 0.6,
  "palette": 8,
  "coloring": 18
}

Use EXPORT SUMMARY in the engine to capture the full reproduction key for any view state.

Usage

No install. No dependencies. Open index.html in any modern browser.

• Scroll — zoom
• Drag — pan
• Click — set Julia seed at cursor position
• R — reset view

Dependencies

| Library | Version | License | |---------|---------|---------| | WebGL2 / GLSL 300es | — | Open standard (Khronos) | | Google Fonts — Space Mono, Syne | — | SIL OFL 1.1 |

No npm packages. No build step. No bundler.

uuon-clouud Integration

The planned API layer exposes this engine as a callable service:

• POST /api/v1/fractal/render — PNG from parameter seed
• POST /api/v1/fractal/field — Float32Array escape values (heightmap/mask)
• POST /api/v1/fractal/seed — Store named seed with provenance
• GET /api/v1/fractal/seed/:id — Retrieve named seed

See api/README.md for the full specification and next-session starting point.

Known Limitations at v1.0.0

| Limitation | Impact | Resolution | |-----------|--------|------------| | Generator 15 at high power | atan(z.y,z.x)*power can overflow into NaN before guard at p=4.0, large orbit | Clamp phase before rotation in v1.1.0 | | u_pa/u_pb dual role | Warp params and orbit trap point (coloring 4) share the same uniforms — behavior changes silently | Separate uniforms in v1.1.0 | | No CPU pipeline | Server-side rendering requires porting GLSL loop to JS | Build api/lib/pipeline.js | | No seed registry | Named attractors can't be stored server-side | Postgres table after API server | | highp float precision cap | Zoom beyond ~10¹⁰ degrades — WebGL2 highp is ~7 decimal digits | Expected hardware limitation |

License

USAL-1.0 — see LICENSE. Attribution required. AI training use prohibited. Commercial licensing: [email protected]

UUON Foundation Inc. · Phillip Aguilar Ruiz III