Four Hundred Years of Controversy: The Complex Number i as Phase Migration of the Self-Referential Mind-Field on the Mathematical Section — A Sectional Analysis of the Ontological Dispute over the Imaginary Unit within the Framework of Yuanxian Theor

Since the birth of the concept of imaginary numbers in the sixteenth century, the ontological controversy surrounding the complex number \\(i\\) has run through four centuries of mathematics, physics and cognitive science. From Cardano’s “sophistic quantity”, through Euler’s symbolic establishment, Gauss’s geometrisation of the complex plane, Schrödinger’s embedding of \\(i\\) at the core of quantum mechanics, to the 2021 experimental demonstration that complex numbers are necessary and subsequent work proposing equivalent real-number versions—the controversy has never received a unified ontological explanation. Drawing on the core framework of Yuanxian Theory—the 64-dimensional compact torus \\(T^{64}\\) as topological ontology, the self-referential mind-field \\(\\Psi^{*}\\) as modal of existence, and sectional projection as mechanism of manifestation—this paper advances the following central claims: - The complex number \\(i\\) is not an artificially invented computational tool, but a phase-migration marker of the self-referential mind-field on the real-mathematical section.- The rotational, phase and self-reflexive properties of \\(i\\) are essentially the mathematical traces of “self-referential fold-back” that necessarily appear when a self-referential structure is projected onto a lower-dimensional real space.- “Complex numbers are necessary” and “real numbers also work” are not contradictory—they are manifestations of \\(\\Psi^{*}\\) at different reading resolutions on the mathematical section. --- 自16世纪虚数概念诞生以来,复数 \\(i\\) 的本体论争议贯穿数学、物理、认知科学四百年。从卡尔达诺的“诡辩量”,到欧拉的符号确立,到高斯的复平面几何化,到薛定谔方程将 \\(i\\) 嵌入量子力学核心,再到2021年实验确认复数必需与后续研究提出等价的实数版本——这场争议始终未得到本体层面的统一解释。 本文基于元宪理论的核心框架——以 \\(T^{64}\\) 六十四维紧致环面为拓扑本体、以自指心场 \\(\\Psi^{*}\\) 为存在模态、以截面投影为显影机制——提出核心论点: - 复数 \\(i\\) 不是人为发明的计算工具,而是自指心场在实数数学截面上的相位迁移标记。- \\(i\\) 的旋转、相位、自反特性,本质是自指结构投射到低维实数空间时必然出现的“自指折返”数理痕迹。- “复数必需”和“实数也行”不是矛盾——它们是 \\(\\Psi^{*}\\) 在数学截面上不同读数分辨率的显影。