Computable Coordinate System Objects: Theory and Applications Article Swipe

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Pan Guojun ·

YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.5281/zenodo.17850779

— We present a unified computational framework for geometry and physics based on the concept of Computable Coordinate Systems. The central idea is to elevate coordinate systems from passive references to first-class algebraic objects, denoted as coord, which support intuitive operations such as multiplication (∗) and division (∕). This algebraic approach replaces cumbersome matrix and tensor calculus with a natural and efficient formalism for hierarchical transformations. The framework extends naturally into differential geometry through the Intrinsic Gradient OperatorGμ=ΔcΔμ∣c-frame,Gμ=ΔμΔcc-frame,which measures the variation of a frame field within its own coordinate system. Curvature is then obtained directly via the Lie bracket [Gu,Gv][Gu,Gv], with a metric normalization that ensures coordinate invariance. Beyond geometry, we unify complex frame transformations, Fourier transforms, and conformal mappings within the same algebraic structure, providing a geometric foundation for path integrals, gauge theories, and quantum field theory. Numerical experiments on canonical surfaces confirm machine-precision accuracy and a 275% speedup over classical methods. This work thus bridges abstract mathematics with practical computation, offering a unified language for graphics, robotics, simulation, and theoretical physics.

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Algebra over a field Coordinate system Mathematics Differential geometry Algebraic number Conformal map Differential algebraic geometry Matrix multiplication Tensor calculus Algebraic geometry Pure mathematics Tensor algebra Gauge theory Field (mathematics) Manifold (fluid mechanics) Multivector Calculus (dental) Tensor field Computer science Curvature Affine variety Normalization (sociology) Vector field Frame of reference

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Type: preprint
Language: en
Landing Page: https://doi.org/10.5281/zenodo.17850779
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DOI: https://doi.org/10.5281/zenodo.17850779

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Title: Computable Coordinate System Objects: Theory and Applications

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Publication year: 2025

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Publication date: 2025-12-08

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Authors: Pan Guojun

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Concepts: Algebra over a field, Coordinate system, Mathematics, Differential geometry, Algebraic number, Conformal map, Differential algebraic geometry, Matrix multiplication, Tensor calculus, Algebraic geometry, Pure mathematics, Tensor algebra, Gauge theory, Field (mathematics), Manifold (fluid mechanics), Multivector, Calculus (dental), Tensor field, Computer science, Curvature, Affine variety, Normalization (sociology), Vector field, Frame of reference

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