Superlinear efficiency — Geometric invariance — Constant‑memory transport
This page outlines the core computational laws governing the fractal streaming primitive. These laws explain why the operator exhibits superlinear efficiency, geometric invariance, and stable long‑range transport across both symbolic and physical domains. Together, they form the scientific foundation for treating the primitive as a reusable computational substrate rather than a task‑specific algorithm.
The primitive demonstrates a reproducible superlinear efficiency law as spatial resolution increases. While the baseline implementation remains strictly linear and small, the primitive’s curve bends upward—a signature of coherent transport and directional amplification rather than brute‑force computation.
| Resolution | Efficiency |
|---|---|
| 128×128 | 6.164× |
| 256×256 | 6.689× |
| 512×512 | 6.925× |
| 1024×1024 | 7.039× |
| 2048×2048 | 7.090× |
The baseline remains effectively linear and tiny. The primitive’s efficiency increases with scale, revealing a transport mechanism that compounds rather than degrades as resolution grows.
From controlled GPU‑normalized measurements:
| Metric | Value |
|---|---|
| Fractal efficiency | 0.96752 |
| Baseline efficiency | 0.0020747 |
| Efficiency gain | 46,535% |
This is the strongest normalized scaling signature in the dataset. It shows that the primitive is not merely fast—it is structurally efficient.
The scaling behavior reflects a set of underlying computational laws:
Beyond efficiency, the primitive exhibits a coherent geometric structure. Its behavior is not heuristic; it is governed by a stable law. The invariance suite and geometric fingerprints show that the operator maintains structure across transformations, hardware conditions, and directional inputs.
| Property | Behavior |
|---|---|
| Symmetry | consistent under mirrored configurations |
| Isotropy | uniform performance across directions |
| Angle‑independence | stable transport across arbitrary angles |
| Rotational invariance | structure persists under rotation |
| Directional tunability | directionality can be modulated without breaking coherence |
| Multi‑angle transport law | stable across compound directional inputs |
| Prefix invariance | early‑state transformations do not affect final‑state efficiency |
| Translation invariance | shifting the input space preserves structure |
| Stream vs in‑memory | identical geometric fingerprints |
| Scaling exponent | α ≈ 0.9539 |
These invariances demonstrate that the primitive is governed by a stable geometric law, not an emergent artifact of GPU scheduling. The structure is consistent, reproducible, and mathematically constrained—the hallmark of a genuine computational primitive.
A consolidated research artifact containing the full scaling law analysis, geometric invariance suite, and reproducibility notes will be made available here. This document will serve as the canonical reference for the primitive’s computational laws.
The downloadable PDF artifact will appear here once finalized.