Harmodynamics III: The Baryonic Tully-Fisher Relation as an Emergent Law of the Vacuum

Abstract The Baryonic Tully-Fisher Relation (BTFR) describes a tight empirical power-law corre- lation between the total baryonic mass of a galaxy and its asymptotic rotation velocity (Mb ∝ V 4 ). Within the standard ΛCDM paradigm, this relation presents a significant chal- lenge, requiring fine-tuned feedback mechanisms to reconcile the dynamics of dark matter halos with visible matter. In this work, we demonstrate that the BTFR is a natural and inevitable consequence of the Harmodynamic framework. Building on the observational evidence from our pre- vious work (Paper II), which identified a universal critical acceleration scale of acore ≈ 5.15 × 10−11 m/s2 at the onset of the scalar field condensation, we analytically derive the M ∝ V 4 scaling law. We show that the normalization of this relation is determined solely by the gravitational constant G and the cosmic acceleration scale. This result unifies galac- tic dynamics with the thermodynamic properties of the vacuum, rendering the BTFR a fundamental law rather than an evolutionary acciden