A Testable Tully–Fisher Relation from a Restricted Einstein–Aether Subspace

We study a restricted subspace of Einstein–Aether theory in which the self-acceleration term is absent ([Formula: see text]). After integration by parts, the theory is equivalent to a vector–tensor model with a non-minimal Ricci coupling. In the weak-field, quasi-static limit, and after imposing the gravitational-wave speed constraint [Formula: see text], the modified Poisson equation can be written in the AQUAL form with an effective acceleration scale [Formula: see text]. This scale is set by the cosmological Hubble parameter and depends on the coupling constants only through the combination [Formula: see text]. In the deep-MOND regime the baryonic Tully–Fisher relation takes the form [Formula: see text]. A Markov-chain Monte Carlo analysis of the SPARC galaxy sample shows that the parameter combination [Formula: see text] is constrained to be close to unity, so that the Tully–Fisher normalization is nearly universal. The same subspace can generate an effective dark-energy component with equation of state [Formula: see text], but the stronger PPN [Formula: see text] bound restricts the available coupling strength so that the minimal model cannot simultaneously explain the observed cosmic acceleration. Additional freedoms beyond the restricted subspace are therefore required for a full unified scenario.