Bradley Oscillation Field Theory: Derivation of the Radial Acceleration Relation and Galactic Rotation Curves

We present a zero-parameter continuum mechanics derivation of flat galactic rotation curves, the Baryonic Tully-Fisher Relation (BTFR), and the Radial Acceleration Relation (RAR) operating strictly on flat, uncurved Minkowski spacetime (ημν=diag(-1,1,1,1)). By treating the scalar vacuum as a physical wave-transmission substrate with exponential impedance (Z(Φ)=e^Φ), gravitational propagation splits into a three-dimensional spherical expanding wave and a trapped two-dimensional cylindrical wave floor. Reinterpreting the Hubble constant (H₀) as the intrinsic harmonic attenuation coefficient of the scalar medium, we derive the kinematic acceleration floor from first principles as aB = c₀ H₀/2π = 1.0422 × 10⁻¹⁰ m/s². This establishes the Rene Limit rR = √GMb/aB, where a critical scalar impedance mismatch induces Total Internal Reflection, trapping outer disk stars within a stable, two-dimensional cylindrical potential well. Step-by-step evaluations demonstrate that local planetary orbits (r rR) remain $>98.3%$ Keplerian (4.78 km/s), while macro galactic systems (NGC 3198, r rR) yield flat rotation speeds (150.11 km/s) matching SPARC observations without non-baryonic dark matter halos or empirical interpolation functions.