Finite-resolution measurement induces topological curvature defects in spacetime

Abstract: We show that regularizing $(2+1)$-dimensional Minkowski spacetime with a finite-resolution Gaussian probe, analogous to Weyl-Heisenberg (Gabor) signal analysis and related quantization, induces a curved geometry with a topological defect. The regularized metric replaces $r^2$ by $r^2+σ^2$ in the angular part, where $σ$ is the resolution scale from the width of the Gaussian probe. The resulting Gaussian curvature integrates to $-2π$, independently of $σ$, and including the boundary contribution, yields Euler characteristic $χ=0$, corresponding to a punctured plane. This curvature defines an effective stress-energy source with total energy $E_{\text{eff}}=-1/(4G)$, universal and $σ$-independent. Spatial slices embed isometrically as helicoids, and geodesics exhibit a characteristic swirling motion. These results show that finite spatial resolution measurement does not merely smooth singularities but imprints topological defects with fixed physical consequences, suggesting that observational limitations fundamentally shape spacetime geometry. We show how our Gabor regularisation is extendable to $(3+1)$ Minkowski space-time.