The post has been translated automatically. Original language: Russian
Authors: Oleg S. Basargin
Affiliation: Sfiralium research foundation
Keywords: Sfiral Neural Networks, Split-Complex Numbers, Topological Protection, Phase Transition, Hypercomplex Algebra, Noise Immunity.
Abstract
Modern deep learning architectures, primarily based on real and complex arithmetic, face fundamental limitations in handling high-dimensional noise, phase decoherence, and topological instability. In this paper, we introduce the Sfiral Artificial Neuron (SIN) and its fractal extension (FSIN), a novel neural architecture grounded in the algebra of split-complex (hyperbolic) numbers ⅅ. Unlike traditional Möbius strip or Archimedean spiral models, the Sfiral topology is defined by two mirror-antisymmetric coils connected by an S-shaped phase transition loop. We mathematically formalize the S-loop as a topological filter operating near the null-cone (∣Z∣2 = 0) of the split-complex plane, providing inherent noise immunity and phase stabilization. Furthermore, we propose a fractal embedding mechanism that allows multi-scale temporal and spatial feature extraction without exponential parameter growth. The Sfiral framework offers a new paradigm for topologically protected quantum-inspired computing and robust AI systems.
Basargin, O. S. (2026). Hypercomplex Phase Dynamics and Topological Protection in Split-Complex Space. Zenodo. https://doi.org/10.5281/zenodo.21208983
Software application: https://sfiraliumlab-ux.github.io/Sfiralium-Core/#demo
Authors: Oleg S. Basargin
Affiliation: Sfiralium research foundation
Keywords: Sfiral Neural Networks, Split-Complex Numbers, Topological Protection, Phase Transition, Hypercomplex Algebra, Noise Immunity.
Abstract
Modern deep learning architectures, primarily based on real and complex arithmetic, face fundamental limitations in handling high-dimensional noise, phase decoherence, and topological instability. In this paper, we introduce the Sfiral Artificial Neuron (SIN) and its fractal extension (FSIN), a novel neural architecture grounded in the algebra of split-complex (hyperbolic) numbers ⅅ. Unlike traditional Möbius strip or Archimedean spiral models, the Sfiral topology is defined by two mirror-antisymmetric coils connected by an S-shaped phase transition loop. We mathematically formalize the S-loop as a topological filter operating near the null-cone (∣Z∣2 = 0) of the split-complex plane, providing inherent noise immunity and phase stabilization. Furthermore, we propose a fractal embedding mechanism that allows multi-scale temporal and spatial feature extraction without exponential parameter growth. The Sfiral framework offers a new paradigm for topologically protected quantum-inspired computing and robust AI systems.
Басаргин, О. С. (2026). Hypercomplex Phase Dynamics and Topological Protection in Split-Complex Space. Zenodo. https://doi.org/10.5281/zenodo.21208983
Приложение: https://sfiraliumlab-ux.github.io/Sfiralium-Core/#demo
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