Density Dependence of the Gravitational Constant: Compatibility Test of the ExB-T Prediction G ∝ ρ^(−1/2) with SPARC Rotation Curves

We test the density dependence of the gravitational constant: whether G varies with ambient density according to G ∝ ρ^(−1/2). The ExB-T geometric theory treats G as a fluid stiffness coefficient: a black hole is regarded as a closed fluid region, and the stiffness coefficient is related to the square root of density, giving an exponent of −1/2. This expression yields G from geometric quantities, differing from the CODATA 2018 value by 0.9%; this difference is consistent with the density dependence predicted by the theory [29]. Here we use 135 galaxies and 3014 data points from the Spitzer Photometry and Accurate Rotation Curves (SPARC) database to test whether G_eff/G varies with baryonic acceleration g_b as a power law with exponent −1/2 on galactic scales. The results are threefold. First, the data reject the hypothesis that G is constant. Five complementary tests consistently reject independence of G_eff/G from g_b: Spearman ρ = −0.8578 (galaxy-level 95% confidence interval, CI, [−0.8884, −0.8154]), F-test p < 10^−15, strictly monotonic decrease across eight bins, and ΔBIC = 3320.66. Second, the variation of G is consistent with G ∝ ρ^(−1/2). A power-law fit to the full data gives n = −0.4799, with a galaxy-level 95% CI [−0.5431, −0.4050] that includes −1/2; the exponent from theoretical bin centers is −0.4355, while that from the actual data medians across eight bins is −0.6468; 120/135 galaxies (89%, without screening) show the same sign. Under the baseline full sample and the nominal SPARC mass-to-light ratio (M/L), the −1/2 prediction is compatible with the data. Third, the boundaries are explicit. The gas-dominated subsample gives n = −0.7033 (galaxy-level CI [−0.8221, −0.5397]), which excludes −1/2; the extreme low M/L case Υ = 0.5 gives n = −0.4047 (CI [−0.4907, −0.3304]), which excludes −1/2; the extreme high M/L case Υ = 2.0 gives n = −0.5730 (CI [−0.6413, −0.4797]), which includes −1/2 but is steeper. Binning robustness shows that the n values from four alternative binning schemes span 0.10, and including the eight-bin actual median the total span across six schemes is 0.23; residuals are systematically correlated with f_gas, log₁₀V_flat, and log₁₀g_b. The −1/2 exponent is compatible with the data under the baseline full sample and nominal M/L, but its range of applicability is delimited by M/L calibration, sample selection, and binning scheme. G_eff/G = g_obs/g_bar is the radial acceleration relation (RAR) in another notation. The RAR can arise in dark matter, MOND (Modified Newtonian Dynamics), and ExB-T frameworks alike; what this study tests is the ExB-T mechanistic interpretation: that G itself varies with density. The conclusion is: SPARC data reject a constant G, and G ∝ ρ^(−1/2) is compatible with the data under the baseline full sample and nominal M/L; the boundaries of this compatibility are defined by the gas-dominated subsample, extreme M/L, binning scheme, and systematic residual correlations.