Topological Asymmetry and Affine Invariance of N-Dimensional Discrete Sets: A Generalized Measure-Theoretic Framework

This paper establishes a generalized measure-theoretic framework to quantifyspatial asymmetry and internal topological divergence within arbitrary 𝑁-dimensional discrete Euclidean spaces ℝ^𝑁. We formalize the Generalized 𝑁-Dimensional Vector Drift Operator 𝓓(𝕊), defined as the exact algebraicdifferential between the discrete mathematical expectation of empirical Diracmeasures 𝛿 and the continuous hypervolumetric expectation of the dynamicallyinduced bounding convex polytope (evaluated via the 𝑁-dimensional Lebesguemeasure λ_𝑁). To mathematically neutralize the determinant of arbitrary affinescaling matrices and guarantee dimensional consistency, the Euclidean norm ofthis structural drift is rigorously normalized against the principal 𝑁-th root of thetopological scale. This yields the strictly dimensionless Topologically InvariantDivergence Coefficient 𝒦̃_𝑁(𝕊). The universal stability of this mathematicalfunctional is analytically and numerically verified across an exhaustive spectrumof multidimensional states, spanning absolute stasis (𝑁-dimensional hypercubiclattice), bounded non-monotonic variance (high-dimensional hyperchaos),asymptotic equilibrium (𝑁-dimensional prime integer lattice), and severe spatialdeformations (anisotropic algebraic singularity). Ultimately, the frameworkdelivers a fundamental, coordinate-independent, and scale-invariant geometrictopology tool, proving that pure structural asymmetry can be exactly quantifiedacross arbitrary dimensions without succumbing to metric degeneration.