The Generative Origin of the Imaginary Unit
Year: 2025
Type: article
Abstract: This work introduces a generative foundation for the imaginary unit i. We show that i is not an arbitrary algebraic invention, but the unique quarter-cycle operator emerging from a four-phase generative geometry. The paper formalizes the generative disc, the cycle operator C: {0,1,2,3}, the continuous anomaly-phase operator Aθ, and the discrete depth operator Dn. Together, these form a hybrid state space that couples smooth dynamics with discrete Feigenbaum scaling (α≈2.5029, δ≈4.6692) under the regulatory irrationality of φ (the golden ratio). We prove that no 2-phase or 3-phase structure can generate the algebraic properties of i, and that the four-phase cycle (Z4) is the minimal, self-consistent geometry producing the full complex exponential sequence (e^{iπ/2}=i, e^{iπ}=-1, e^{3iπ/2}=-i, e^{2iπ}=1). A short remark connects this minimal structure to higher-dimensional extensions such as quaternions. The resulting generative algebra provides a discrete–continuous hybrid framework for complex numbers, dynamical systems, operator theory, and nonlinear oscillatory processes. It reframes the imaginary unit as an emergent generative operator rather than an abstract symbolic constant.
Author: Sikora, Artur
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7105741883
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