Geometry of Deformed Cellular Spaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914367438585856 |
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| author | Barak, Shlomo Salman, George |
| author_facet | Barak, Shlomo Salman, George |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03285 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometry of Deformed Cellular Spaces Barak, Shlomo Salman, George General Mathematics 53C20, 52C99, 53B20, 05C12 F.2.2; G.1.0 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. |
| title | Geometry of Deformed Cellular Spaces |
| topic | General Mathematics 53C20, 52C99, 53B20, 05C12 F.2.2; G.1.0 |
| url | https://arxiv.org/abs/2603.03285 |