Vortex-Induced Information Frequency Compression: A Natural Manifestation——Mathematical Description from Logarithmic Spiral to Multi-Layer Nesting

Based on the framework of Primitive Space Theory and World Quantum Theory, this paper explores the natural manifestation of information frequency compression within vortex structures. By introducing the logarithmic spiral as the geometric prototype of vortex expansion, combined with the rotational expression of Euler‘s formula, a unified mathematical form of “rotation with simultaneous radius scaling” is constructed. This form is further extended to multi-layer nested structures, yielding a quantitative relationship describing how information frequency decreases layer by layer and wavelength increases layer by layer when propagating outward from the vortex core. This paper argues that information frequency compression in vortices is not an externally imposed “design” but a geometric-dynamic necessity naturally emerging from the vortex’s need to maintain coherence, cross-scale propagation, and identifiability. This mechanism provides a computable mathematical framework for understanding the “recurrence of identical topological structures at different scales” and offers an intuitive geometric interpretation of the norm relation K2=∑xn2 in Hypothesis 2 of Primitive Space Theory. Keywords: Vortex; Information Frequency Compression; Logarithmic Spiral; Multi-Layer Nesting; Cross-Scale Self-Similarity; Primitive Space Theory