Geometric Origin of Dark Matter in the S³ Noncommutative Geometry Grand Unified Theory

Within the framework of the S³ noncommutative geometry grand unified theory, this paper demonstrates that the astronomically observed "dark matter" effect does not originate from unknown elementary particles, but is an intrinsic geometric effect of the noncommutative spacetime Sq³ R at galactic scales. Based on the spectral action principle and the scale self-duality axiom, we derive the effective energy-momentum tensor induced by noncommutative geometry from first principles, whose energy density strictly follows the form ₃₌ (r) 1/r². This density profile naturally explains the flatness of galaxy rotation curves, and predicts a linear proportionality between the asymptotic rotation velocity V₅₋₀ₓ and the optical radius R₂₅ of galaxies — a relation that has been precisely verified with high statistical significance by the observational data of the SPARC survey. All geometric constants in the theory are uniquely fixed by fundamental scales (the Planck length lₚ and the Hubble radius RH), without any free parameters. The theory naturally avoids the small-scale crises of the model, and smoothly transitions to the classical limit at the solar system scale. This study marks the first unified description of dark matter, dark energy, and baryonic matter within a single geometric framework, extending the theory from the Planck scale to the Hubble scale.