Geometry of Deformed Cellular Spaces

TL;DR

This paper introduces an adaptive, cell-count based geometry framework that measures distances and curvature without relying on traditional shapes or embeddings, linking discrete cellular data to continuum geometric concepts.

Contribution

It develops a micro-agnostic, operational approach to geometry on cellular spaces, establishing geodesic properties, stability, and a conformal metric connection to smooth geometry.

Findings

Count metric on complexes is geodesic

Flatness on uniform lattices proven

Distances and curvature estimators are stable under perturbations

Abstract

We present an adaptive geometry in which the yardstick co-deforms with space itself, formulated on cellular spaces where length is a count: distances are shortest cell-crossing counts. No cell shape, angles, or embedding are assumed; the framework is deliberately micro-agnostic. Curvature and deformation are inferred operationally by comparing a measured radius to a radius reconstructed from boundary/area/volume counts; the linear dimension of a cell serves as the single universal unit of length, yielding unified small-ball/small-sphere estimators in 2D/3D/4D. We prove that the count metric on locally finite complexes is geodesic, show flatness on uniform lattices, and establish stability of distances and curvature estimators under small local perturbations. As a bridge to the smooth setting, a line-density field induces a conformal metric $g = e^{2u} g_0$ that reproduces the same operational quantities. We outline a Ricci-like construction from directional slices and give a spherically symmetric illustrative example consistent with Schwarzschild-type spatial behavior. Overall, the model provides an intrinsic, micro-agnostic calculus linking discrete measurements to continuum notions with guarantees, including Gromov--Hausdorff control under mild regularity assumptions.