Batalin–Vilkovisky Functional Renormalization Group for Quantum Distortion Gravity: Regulator-Induced Symmetry Breaking, Anomaly Status and Fixed-Point Existence

Quantum Distortion Gravity (QDG) is a metric-affine theory with gauged Weyl symmetry. We set up its Batalin-Vilkovisky (BV) formulation and combine it with the Wetterich functional renormalization group. The BRST transformations of \( Diff Weyl \) (with local Lorentz transformations for fermions) are written out, their nilpotency and the classical master equation are established, and the Faddeev-Popov action in unitary gauge is shown to have a field-independent Weyl-ghost determinant. For a regulator that does not respect the BRST symmetry we derive the modified Slavnov-Taylor identity with an explicit regulator term \( K_k \) and show that the Wetterich flow preserves it; the resulting "third equation" for \( ∂_t K_k \) is an integrability condition and a diagnostic of truncations, not an independent dynamical law. The 60 non-axial components of the Weyl-covariant distortion are auxiliary with constant propagator, so they form an ultralocal Gaussian block that contributes to no curvature-dependent coefficient; the axial vector is a Proca field and does contribute. The anomaly coefficients of the matter content are \( (c,a)=(11/30,\,53/120) \) for nf=3 Dirac fermions and two Proca fields; the graviton has no such pair, and the anomaly of the gauged Weyl symmetry is open. With the Einstein-Hilbert flow and minimally coupled matter, the QDG-like content has a non-Gaussian fixed point for every nf<10, regulator-robust up to \( n_f9 \), but at nf=3 it lies at g*=2.73, \( λ_*=-0.32 \), outside the range where the truncation is reliable; Weyl-gauged operators, gauge dependence and non-minimal couplings are not included. QDG remains a proposal; the status of each result is stated throughout.