Computable Coordinate System Objects: Theory and Applications

We introduce a new paradigm that unifies geometry, physics, and computation by promoting coordinate systems to primary algebraic objects (coord). This framework replaces traditional tensor calculus with intuitive operations like multiplication (*) for composition and division (/) for inversion, enabling efficient hierarchical transformations and precise curvature computation. The geometric core is the Intrinsic Gradient Operator G_μ = (Δc/Δμ) |c-frame, which measures how a frame field varies within itself. Curvature is derived intrinsically via the Lie bracket Gᵤ, Gᵥ with metric normalization ensuring coordinate invariance. This leads to the Complex Frame Unification Theory (CFUT), which geometrizes all fundamental interactions through a U (3) complex frame field U (x). Its dynamics are governed by a unified field equation—the "Christmas Equation"—with rigorous real/imaginary decomposition: MP² / 2 * Ĝ_μνU + (λ/ (32π²) ) * ∇̂_ (μ K̄_ν) U = T̂^ (top) _μνU + T̂^ (mat) _μν where: Ĝ_μν (real part) captures spacetime geometry (gravity) K̄_μ (imaginary part) represents topological Chern-Simons currents T̂^ (top) _μν sources topological phenomena (dark matter) T̂^ (mat) _μν sources conventional matter/energy Key Predictions & Validations: Dark Matter: ~10 TeV topological vortex (ΩDM ≈ 27. 1%), detectable by DARWIN by ~2035 Nuclear Fusion: 12–18% confinement time enhancement in ITER via Chern-Simons stabilization Gravitational Waves: Frequency-dependent polarization asymmetry (δ ∝ fGW), testable by LISA-Taiji Computational: Machine-precision curvature (ε < 1e-15), O (n⁴) → O (n²) complexity reduction, 3. 75× speedup The theory is validated across 19 orders of magnitude and bridges abstract mathematics with practical computation. The 2025–2035 experimental campaign will provide definitive tests.