Pressure-Mediated Gravity III Rotation Curve Fits and the Radial Acceleration Relation for 175 SPARC Disk Galaxies

We present the first complete, quantitative application of Pressure-Mediated Gravity (PMG) to the full Spitzer Photometry and Accurate Rotation Curves (SPARC) database of 175 late-type disk galaxies. PMG is an emergent-gravity framework in which gravitational dynamics arise from pressure gradients in a superfluid vacuum governed by a logarithmic Gross-Pitaevskii Lagrangian (Jerrow 2026a, PMG v6). The interpolation function mu_PMG(x) = x/sqrt(1+x^2) and the acceleration scale a0 = c*H0/2pi = 1.051e-10 m/s^2 are both derived from first principles rather than fitted empirically. This paper derives the PMG-specific nu_PMG(y) function as the exact explicit solution of the PMG v6 mu-form implicit equation mu(g_obs/a0)*g_obs = g_bar, obtained by solving the quartic t^4 - y^2*t^2 - y^2 = 0. This function is distinct from the standard MOND simple nu(y) function by up to 21% in the transition regime and is verified numerically to machine precision. Fitting 175 SPARC galaxies with free stellar mass-to-light ratios Upsilon_disk in [0.01, 10.0] M_sun/L_sun, we find: - Median chi^2_nu = 2.57 (baseline); 2.29 marginalising over distance (+/-10%) and inclination (+/-3 deg) uncertainties - 73/175 galaxies (42%) with chi^2_nu < 2 (good fits) - 115/175 galaxies (66%) with chi^2_nu < 5 (acceptable fits) - Median Upsilon_disk = 0.63 M_sun/L_sun (physically consistent with stellar population synthesis predictions) - BTFR slope = 4.00 +/- 0.03 — exact confirmation of the PMG parameter-free prediction M_bar proportional to V_flat^4 - RAR scatter of 0.133 dex, matching the observed total scatter An independent scan over a0 yields a kinematic measurement a0_kin = 1.135e-10 m/s^2, only 8% above the PMG-theoretical value and sitting between the PMG theory value and the empirical MOND scale (1.2e-10 m/s^2), consistent with both at <1 sigma. This is the first kinematic measurement of a0 within the PMG framework. The 60 poor-fit galaxies are predominantly irregular or kinematically disturbed systems; no systematic over-prediction of baryonic acceleration is found. Three open challenges — environmental dependence of a0, Bullet Cluster lensing, and CMB acoustic peaks — are identified as Papers IV, V, and VI of the PMG research programme. Data: SPARC database (Lelli, McGaugh & Schombert 2016, AJ 152 157). Code: Python pipeline included in this deposit.