From Scratch II: An Empirical Test of the Flat-Universe Assumption

This paper evaluates the spatial curvature parameter Ωk using only late-time observations at red- shifts z ≤ 2.5. Cosmic chronometers provide direct measurements of H(z), and BAO contribute only dimensionless ratios that do not introduce an absolute scale. A Gaussian-Process reconstruction of H(z) combined with Monte Carlo propagation through curvature-dependent distance relations shows that late-time data constrain Ωk only weakly (σ ≈ 0.01). Because curvature sensitivity arises primarily from z > 2, where no direct expansion-rate observations exist, current late-time measurements do not empirically determine whether the Universe is open, flat, or closed.