Geometry of Deformed Cellular Spaces

We develop an operational, measurement-first framework for the geometry of locally finite cell complexes, in which length is defined as a count of face crossings, and curvature is read off from the discrepancy between a measured radius and a radius reconstructed from boundary, area, or volume counts using the same yardstick. We prove that the count metric is geodesic on every locally finite complex, and we introduce a unified small-ball/small-sphere curvature estimator that is valid in dimensions two through four with a single closed-form expression. By comparison with the standard small-ball volume expansion of a smooth conformal metric g=e2ug0, we establish a quantitative identification theorem with explicit rate O(a/r+r), which optimizes to O(a) at r≍a. We extend the construction to directional (sectional) estimators via Fermi tubes around geodesic two-slices, assemble the curvature operator, Ricci tensor, and scalar curvature in three dimensions, and prove a measured Gromov–Hausdorff convergence theorem for the rescaled count metric. All hypotheses are verified explicitly on Voronoi complexes of conformal metrics. Throughout, we are explicit that the discrete construction is interpreted via, and its asymptotic validity is established by comparison with, the smooth Riemannian theory; the contribution is the unified counts-only protocol with rigorous convergence rates, not a reformulation of curvature itself.