Isotropic Equivalence of STVG--MOG and $\Lambda$CDM and Its Breakdown in Large--Scale Anisotropic Cosmological Observables

We show that Scalar-Tensor-Vector Gravity (STVG-MOG) is observationally equivalent to the standard model $\Lambda$CDM cosmological model for all probes that depend on isotropic and linear gravitational dynamics, including galaxy rotation curves, cluster lensing, the linear matter power spectrum P(k), $\sigma_8$, baryon acoustic oscillations, and the cosmic microwave background (CMB). This degeneracy arises from the scale-dependent effective gravitational coupling $G_{\mathrm{eff}}$, which ensures identical background evolution, transfer functions, and linear growth. Consequently, all early-universe, low and intermediate scale cosmological observables are equally well described by STVG-MOG without invoking non-baryonic dark matter. We argue that the equivalence implies that isotropic cosmological data alone cannot establish the physical existence of dark matter. The degeneracy is broken only by observables sensitive to large-scale, anisotropic gravitational response. In particular, recent measurements of enhanced radio-galaxy and quasar number-count dipoles at gigaparsec scales probe a regime where $G_{\mathrm{eff}}$ departs from its $\Lambda$CDM limit, allowing STVG-MOG to generate anisotropic bulk flows, while preserving consistency with all isotropic constraints. These observations provide a concrete pathway for empirically distinguishing modified gravity from particle dark matter.