A Regime-Dependent Sign Reversal in Galactic Variance Structure at the Cosmic Matter–Dark Energy Transition

We present a variance-geometry diagnostic for analyzing the internal structure of galaxy scaling relations using the eigenvalue organization of covariance matrices. The method introduces the structure index \lambda_{\mathrm{eff}}, which quantifies how variance is distributed across dominant and sub-dominant modes, providing information beyond traditional slope and scatter analyses.

Applying this diagnostic to galaxy samples spanning matter-dominated and dark-energy-dominated epochs reveals a regime-dependent correlation between \lambda_{\mathrm{eff}} and the cosmological density ratio \Omega_\Lambda/\Omega_m. The correlation reverses sign near the cosmic coincidence point \Omega_\Lambda \approx \Omega_m, and a global minimum in \lambda_{\mathrm{eff}} is observed at z \approx 0.25, close to this transition. This minimum lies below, but near, an empirically identified structural corridor previously associated with variance integration.

These results reconcile previously contradictory trends reported in different galaxy samples by demonstrating that they probe opposite sides of a cosmological regime boundary. The findings suggest that the cosmic coincidence marks a statistically special point in the organization of galactic variance, while remaining fully consistent with standard \LambdaCDM cosmology. The framework is presented as a diagnostic and statistical tool, not as evidence for new physical interactions, and generates clear, testable predictions for future datasets.