A Parameter-Free Derivation of the Baryonic Tully–Fisher Relation from a Relativistic Bose–Einstein Condensate of Time

We present a derivation of the Baryonic Tully–Fisher Relation (BTFR) and of the associated Radial Acceleration Relation (RAR) with zero free parameters, within a theoretical framework in which the physical vacuum is modelled as a relativistic Bose–Einstein condensate of a fundamental temporal substrate (Fluid Time Theory, TTF). In this framework, a critical acceleration scale emerges naturally from cosmological constants alone: a₀ = c H₀ / (2π) ≈ 1. 08 × 10⁻¹⁰ m/s², in close agreement with the value fitted empirically by McGaugh et al. (2016). Testing the resulting BTFR vf⁴ = G Mbar c H₀ / (2π) against the SPARC catalogue of 120 well-measured disk galaxies, we find a slope of 0. 266 ± 0. 01 (predicted: 0. 250), an intrinsic scatter of 0. 062 dex, and 82% of galaxies within ±20% of prediction, without any parameter adjusted to galactic data. A phononic screening mechanism, arising from the Bogoliubov dispersion relation of the condensate in strongly tensioned gravitational backgrounds, suppresses the anomalous acceleration in the Solar System by a factor ~ e^ (-6 × 10⁵), making the theory compatible with Cassini spacecraft constraints at a level ~ 3300 times below the experimental bound. We outline three laboratory- and observation-accessible falsification tests: (i) the LSST slope of the BTFR, (ii) a gravitational resonance at νWF = 1. 02 THz detectable by bulk acoustic wave (BAW) resonators or levitated optomechanics, and (iii) a submillimetre absorption feature at ~ 0. 78 THz around neutron stars, accessible to ALMA Band 10. Keywords: Tully–Fisher relation; Radial Acceleration Relation; modified gravity; dark matter; Bose–Einstein condensate; emergent gravity; SPARC; MOND. --- Version 2 (April 17, 2026). See Additional description for changelog.