Topological Asymmetry and Affine Invariance of 3D Discrete Sets: A Generalized Measure-Theoretic Framework
Year: 2026
Type: preprint
Abstract: This paper establishes a generalized measure-theoretic framework to quantify spatial asymmetry and internal topological divergence within three-dimensional discrete metric spaces. We formalize the Generalized 3D Vector Drift Operator D(S), defined as the exact algebraic differential between the discrete mathematical expectation of empirical Dirac measures and the continuous volumetric expectation of the dynamically induced bounding convex polyhedron (evaluated via the 3D Lebesgue measure). To guarantee absolute affine invariance and dimensional consistency, the Euclidean norm of this structural drift is rigorously normalized against the principal cubic root of the topological scale. This yields the dimensionless Topologically Invariant Divergence Coefficient $\widetilde{\mathcal{K}}_{3D}(\mathbb{S})$. The universal stability of this mathematical comparator is analytically and empirically verified across an exhaustive spectrum of spatial states in $\mathbb{R}^3$, spanning absolute stasis (simple cubic lattice), bounded deterministic chaos (Lorenz strange attractor), asymptotic equilibrium (prime integer lattice), and severe spatial singularities (anisotropic prime powers and logarithmic spirals). Ultimately, the framework delivers a strictly deterministic, coordinate-independent, and scale-invariant computational tool for isolating absolute structural phase shifts in complex 3D distributions.Fixed.
Author: Donchenko, Yaroslav
Language: English
Subfield: Computer Vision and Pattern Recognition
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203705572
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