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Geometry of Deformed Cellular Spaces

Shlomo Barak, George Salman

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.

Geometry of Deformed Cellular Spaces

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 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.
Paper Structure (23 sections, 16 theorems, 144 equations, 4 figures, 1 table)

This paper contains 23 sections, 16 theorems, 144 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Intrinsic metric from counts and geodesics
  3. Operational line element and emergent metric
  4. Metric–measure interface and hypotheses
  5. Excess radius δr and the small--circle law
  6. Perimeter.
  7. Area.
  8. Conformal chart.
  9. Unified curvature estimator from counts
  10. Volume expansion and quadrature error.
  11. Bounding the reconstructed radius.
  12. Main difference bound.
  13. Conclusion.
  14. Local versus global curvature; density interpretation and interface with smooth geometry
  15. Curvature tensor from directional slices
  16. ...and 8 more sections

Key Result

Theorem 2.1

If the adjacency graph is locally finite, then for every $u,v$ there exists a path attaining $d_h(u,v)$. In particular $(\mathrm{Cells},d_h)$ is a geodesic metric space.

Figures (4)

  • Figure 1: Operational meaning of $\delta r$: measured radius $r$ vs. reconstructed radius $r_c$ using the same yardstick.
  • Figure 2: Positive curvature (contraction): $\delta r>0$, $K>0$.
  • Figure 3: Negative curvature (dilation): $\delta r<0$, $K<0$.
  • Figure 5: Fermi tube of thickness $2\tau a$ around the geodesic slice $\Sigma_{ij}$. We sum per-cell weights only inside this tube, then divide by the tube thickness $2\tau a$ to recover an effective slice area. This yields $R_{c,ij}(x;r,a)$ and, via Eq. Eq. (\ref{['eq:Kijhat-def']}), the counts-only estimator $\widehat{K}_{ij,a}(x;r)$ for the sectional curvature $K_g(e_i\wedge e_j)$.

Theorems & Definitions (29)

  • Theorem 2.1: Geodesics on locally finite complexes
  • proof
  • Theorem 3.1: Measured GH limit
  • Theorem 4.1: Small-circle law in 2D
  • proof : Proof.
  • Theorem 5.1: Small-ball identification with rates
  • proof
  • Theorem 6.1: Local identification with scalar curvature
  • Proposition 6.2: Discrete-to-smooth consistency
  • Proposition 6.3: Local/global split in a fixed gauge
  • ...and 19 more