Gravitational Compactness of SPARC Galaxies: Empirical Constraints on Modified Gravity at λ ~ 10⁻⁸

This Paper sets out to measure the dimensionless gravitational compactness: \lambda = \frac{G\,M_{\rm bar}}{R_{\rm eff}\,c^2} for 175 disk galaxies in the SPARC sample, using infrared–derived baryonic masses and 3.6 μm effective radii. This explorations lead to a surprise, and discovered: These λ values cluster very tightly around a median of 2.45\times10^{-8}, roughly log-normal in shape and more than two orders of magnitude below the scale at which many modified-gravity ideas predict observable effects. Next , we asked the question whether a simple global tweak like: V_{\rm model}^2 = V_{\rm bary}^2\Bigl(1 + \kappa\,\lambda^\beta\Bigr) could reproduce galaxy rotation curves ? Fitting \kappa and \beta to 3,387 data points gives an RMSE of 40.8 km/s, only a small improvement over the purely Newtonian result (43.3 km/s) and well behind MOND (∼13 km/s) or NFW dark-matter fits (∼25 km/s). Worse still, the best fit requires \kappa\approx1.8\times10^4, a coupling strength that feels implausibly large; an indication that this global prescription simply misses essential physics. Although this is a null result for λ-only modifications, it still yields something valuable: a new empirical benchmark at \lambda\sim10^{-8}. We hope that future theories - whether they invoke local accelerations, environmental dependencies, or entirely different mechanisms; will honor this scale when tackling the stubborn puzzle of flat galaxy rotation curves.