A Nyquist Criterion for Effective Topology: Theoretical Framework for Resolution-Dependent Structure in Dynamical Triangulations

This technical note introduces a Nyquist-type resolution criterion for effective topology in discrete approaches to quantum gravity, with a specific focus on dynamical triangulations. Building on recent results by van der Duin et al. on scale-dependent Betti numbers, the paper shows that topological phenomena such as pinching, Voronoi cell wrapping, and Delaunay edge bridging are not independent effects but equivalent manifestations of undersampling geometric structure. A single critical scale \ell_{\mathrm{crit}} = \min(L, w, |\kappa|^{-1}) is identified, combining feature size, transverse width, and curvature radius. Faithful representation requires the sampling scale \delta \ll \ell_{\mathrm{crit}}; violations lead to predictable, irreversible artifacts. The framework explains the qualitative difference between Lorentzian (CDT) and Euclidean (EDT) ensembles: CDT lacks local thin necks and remains resolution-stable, while EDT contains fractal necks at all scales, producing immediate topological proliferation. The criterion is diagnostic rather than dynamical: it does not modify quantum gravity models but provides a principled method for distinguishing invariant geometric structure from coarse-graining artifacts. When the sampling scale is identified with the Planck length, the condition reduces to the familiar effective field theory validity bound R \ll M_{\mathrm{Pl}}^2. Conceptually, the approach aligns with Archimedes’ method of exhaustion, treating geometry as something resolved through finite approximation rather than assumed a priori.