Gap Theory V15: Axis-Oriented Dimension Elevation/Reduction Tool

Abstract: Dimension transformation is a core problem in geometric foundations, topological analysis, and related interdisciplinary fields. Traditional dimension elevation/reduction methods often suffer from information loss, irreversibility, and non-standard operations. Based on the axiom system of Gap Theory V3 and the thought experiment of Gap Theory V12, this paper constructs an axis-oriented dimension elevation/reduction tool for 3D and lower Euclidean spaces, featuring action-state separation, step-by-step reversibility, and independent axis operations. The tool abandons global dimension labels and decomposes all transformations into X, Y, Z independent axes, supporting full dimension reduction and half-dimension reduction (0.5D). It archives coordinates and geometric information throughout the process, enabling controllable dimension transformation without information loss, compatible with existing mathematics. This manuscript is currently under review for publication in the Japanese Journal of Mathematics (JJM).