Computable Coordinate System Objects: Theory and Applications

We present a comprehensive theory of computable coordinate systems and frame field combination operators that revolutionizes differential geometry computation. This work integrates three major contributions: Historical-Conceptual Foundation: Traces coordinate systems from ancient civilizations (Chinese Big Dipper, Aboriginal star maps, Babylonian cosmology) through physical paradigm shifts (Ptolemaic → Copernican → Newtonian → Einsteinian → Quantum), revealing coordinate systems as humanity's evolving cognitive framework for understanding spacetime. Mathematical Innovation: Introduces the coord object unifying vectors, matrices, and tensors, alongside frame field combination operators 𝐺𝜇=𝑐(𝑝+Δ𝜇)⋅𝑐(𝑝)−1⋅𝐶(𝑝+Δ𝜇)−1−𝐼⋅𝐶(𝑝)−1 that compute Riemann curvature without Christoffel symbols, reducing complexity from O(n⁶) to O(n³). Physical Applications: Extends the framework to spectral geometry via Fourier transforms, general relativity (tetrad formalism), and computational geometry (shape analysis, mesh processing, robot kinematics). Rigorous proofs establish equivalence with classical Riemannian geometry via Cartan's structure equations. Extensive numerical validation on cone, cylinder, sphere, and torus confirms accuracy to 10−6. This framework bridges discrete computation and continuous mathematics, providing intuitive geometric insight while maintaining mathematical rigor.