Geometric Thermodynamics III:Modified Gravity and Universal Fitting of Galaxy Rotation Curves

This paper presents a comprehensive test of geometric thermodynamics at galactic scales, focusing on the precise fitting of galaxy rotation curves without dark matter. Building upon the scalar-tensor framework developed in Paper II, which predicts a modified gravitational potential $\Phi_{\text{total}}(r) = -\frac{GM}{r}[1 + \Gammapar e^{-r/\lambdaeff}(1 + r/\lambdaeff)]$ with universal parameters $\lambdaeff \approx 4.2$ kpc and $\Gammapar \approx 0.12$, we perform detailed fits to the rotation curve of NGC 3198 and extend the analysis to the SPARC galaxy sample. We provide complete derivations of the circular velocity formula, detailed descriptions of baryonic mass modeling, and exhaustive error analysis. For NGC 3198, we obtain $\chi^2/\text{dof} = 1.1$ with $\lambdaeff = 4.23 \pm 0.28$ kpc and $\Gammapar = 0.121 \pm 0.008$, in excellent agreement with theoretical predictions. Applying the same universal parameters to 175 galaxies from the SPARC sample yields successful fits for 92\% of galaxies with $\chi^2/\text{dof} < 2.0$, demonstrating remarkable universality. We further derive the radial acceleration relation (RAR) from our framework and show it naturally reproduces the observed relation with intrinsic scatter $\sigma_{\text{int}} = 0.11$ dex. All data, fitting procedures, and numerical methods are presented in full detail, establishing geometric thermodynamics as a quantitatively successful alternative to particle dark matter for explaining galactic dynamics.