Does the MOND acceleration scale evolve? Predictions from relativistic MOND and a baryon-budget diagnosis of the reported evolution

Three recent analyses report that the MOND acceleration scale a₀ grows with redshift. This paper asks what relativistic MOND actually predicts for a₀(z), and finds that the reported growth is in 4.7σ conflict with the companion paper of the very series that reports it — and that a single constant correction to the cold-gas budget reproduces the whole signal with no evolution of a₀ at all. THEORY. In the Aether-Scalar-Tensor theory of Skordis and Złośnik [Phys. Rev. Lett. 127, 161302 (2021)] the free function F(𝒴,𝒬) has two arguments, and the quasistatic MOND function is F evaluated at the contemporaneous background value of 𝒬 = A^μ∇_μφ. That value cannot sit exactly at the minimum 𝒬₀ of the cosmological function K, because the departure between them IS the theory's dust-like energy density: setting it to zero deletes the dark matter. So a₀ inherits a redshift dependence, ln[a₀(z)/a₀(0)] = βε₀[(1+z)^(3/n) − 1], where n is the leading power of K about its minimum. Fitting a₀(z) therefore measures the same function that has to reproduce the cosmic microwave background. a₀ is constant if and only if F is additively separable — a condition the authors' own worked example satisfies without comment, and one that the relativistic Khronon theory of Blanchet and Skordis [JCAP 11 (2024) 040] imposes at the level of the action, so that there a₀ is exactly constant. For natural parameters both theories give a₀ constant to better than one per cent at z = 2; reproducing the reported drift needs β ≃ 585, and lunar laser ranging then forces λ_s ≳ 10², eliminating the tracking branch. Because the linear cosmological perturbation equations of the Aether-Scalar-Tensor theory have never been published, the ingredients required are derived here and validated against the published background and Minkowski quadratic actions, which they reproduce exactly. DATA. In MOND the baryonic Tully-Fisher zero point obeys Δb = −Δlog a₀ identically, an identity invariant under the z = 0 mass-to-light anchor. Jeanneau et al. [arXiv:2603.28856] measure no evolution of that zero point (0.00 ± 0.06 dex) while reporting a −0.42 dex shift in the stellar-mass relation; the a₀ evolution of Ciocan et al. [Astron. Astrophys. 709, L16 (2026)] requires −0.30. Moreover the largest single discrepancy in the literature, a₀ = 1.20 against 1.69, arises at essentially zero redshift and tracks the treatment of stellar mass-to-light ratio rather than cosmic time. Taking the cold-gas budget from published scaling relations, one constant gas-recovery fraction reproduces all three quoted values of a₀ to 0.012 dex. The deposit contains the manuscript, the REVTeX source with figures, and scripts that recompute every numerical statement in the paper.