Galactic Kinematics without Dark Matter: Unifying the Radial Acceleration and Baryonic Tully-Fisher Relations via Continuum Field Entropy and the Demise of the 4.0 Slope

The observed discrepancy between the baryonic mass of galaxies and their rotation curves is tra- ditionally attributed to collisionless dark matter halos. Concurrently, standard cosmological models and phenomenological theories (e.g., MOND) rigidly mandate a Baryonic Tully-Fisher velocity slope of S ≡4.0, forcing spatial dependencies to completely cancel out to achieve perfectly flat infinite rotation curves. This paper presents an alternative, unified empirical test utilizing the Continuum Field Entropy (CFE) framework. By deriving the galactic gravitational field as an exact solution to a scalar-tensor variational action operating on a non-linear Cosserat continuum (∇·(Geff(T)∇Φ) = 4πρ), we successfully reproduce the Radial Acceleration Relation (RAR) across 175 SPARC galaxies without dark matter. A full Hierarchical Markov Chain Monte Carlo (MCMC) Bayesian inference isolates an optical retardation factor γopt ≈0.185 and a shear-thickening power-law scaling of α≈2.33, achieving a remarkably tight intrinsic scatter of 0.0965 dex. Applying these exact covariant limits (αβ ≈0.44) to the deep-field BTFR regime, we prove mathematically that the covariant asymptotic limit collapses natively to Mb ∝V∼3.4 f , explicitly rejecting the idealized 4.0 asymptote. We validate this derivation against rigorous Orthogonal Distance Regressions of the SPARC deep-field halos, extracting an em- pirical slope of S ≈3.37. By preserving a residual spatial dependence (R0.21), the CFE framework uniquely predicts the kinematic micro-decay observed at the extreme edges of galactic halos. PrePrint Submitted to The Astrophysical Journal