Topological Asymmetry and Affine Invariance of 2D Discrete Sets: A Generalized Measure-Theoretic Framework

This paper introduces a generalized two-dimensional measure-theoretic frameworkto quantify the spatial asymmetry of discrete subsets on the Euclidean plane ℝ^2.Building upon one-dimensional topological divergence, we formalize the 2D Vector Drift Operator and the dimensionless Topologically Invariant Divergence Coefficient. To preserve absolute affine invariance and neutralize the 𝑆𝑂(2) orthogonal rotation group, the continuous domain is dynamically defined via the convex hull of the subset. Through systematic computational evaluation, we mathematically validate the framework's universality across four fundamental topological classes. The results deterministically isolate absolute structural stasis in a rigid crystalline grid, non-monotonic chaotic variance in a stochastic domain, strict asymptotic equilibrium in Gaussian primes, and exponential structural singularity in the Fibonacci sequence. This constructs a mathematically complete, structurally invariant analytical comparator for evaluating the internal spatial configuration of higher-dimensional discrete sets.