Imaginary Axes Deprivation in Euclidean Space

Version 2: This version corrects errors, improves exposition, and updates terminology and figures to better reflect the article’s scope, while preserving the original preprint’s central thesis and role within the MGQC research program. Abstract The Cartesian formalization of Euclidean space stabilized a three-dimensional structure composed of mutually orthogonal real axes before imaginary quantities acquired a coherent geometric interpretation. When the complex field was later consolidated, internal orientation became mathematically expressible through the imaginary unit, but this enrichment was not extended symmetrically across the three spatial axes of . The article identifies this sequence as a historical and structural asymmetry in the development of , , and . To formulate that asymmetry, the article distinguishes between geometrically effective axes, which determine geometric dimension, and internal orientational parameters, which enrich an axis without generating a new independent spatial direction. On this basis, it introduces coherent dimension as a structural category supplementary to classical geometric dimension and defines three coherent regimes, D1, D2, and D3, corresponding to one, two, and three geometrically effective axes with possible internal orientational enrichment. Within this framework, the complex field may be read not only as a two-dimensional real vector space, but also as a one-axis structure endowed with an internal orientational parameter. This dual readability shows that an increase in parameter count does not necessarily imply an increase in geometrically effective axes. The same principle is then generalized to a coherent three-dimensional model of the form , in which six real parameters are distributed over three geometrically effective axes. From this standpoint, the absence of solutions to equations such as in is interpreted as a structural restriction of the purely scalar regime rather than as a claim extending beyond that regime. The article does not propose a replacement for quaternion algebra and does not challenge classical classification theorems. Its aim is narrower: to clarify an undeveloped axis-wise structural possibility situated between the emergence of internal orientation in and its global non-commutative formalization in . This work forms part of a broader structural investigation into scalar restriction, orientation, and dimensional coherence in classical mathematical systems. This work forms part of the research program Model of General Quasi-Coherence (MGQC). The author publishes under the name Antonio Dominguez-Digat. Earlier records may appear under Antonio Domínguez, Antonio Dominguez, or Antonio Dominguez Digat.