AI--Assisted Exploration: DHOST Theories without Quantum Ghosts

Abstract

We ask whether a local symmetry can organize both classical degeneracy and perturbative stability in DHOST theories. An AI-assisted search finds a candidate in a constant-disformal image of Einstein gravity. We identify it as a field-dependent diffeomorphism plus a vertical translation of a spectator scalar and extend it to every regular first-derivative map $\widetilde g_{\mu\nu}=C(\phi,X)g_{\mu\nu}-D(\phi,X)\nabla_\mu\phi\nabla_\nu\phi$, which pulls the spectator translation back to an exact local gauge redundancy. We derive its closed generator and show that $\mathcal J=C-XC_X+X^2D_X$ controls kinetic invertibility, the generator denominator and the metric-map determinant. This regular ($\mathcal J\neq0$) construction is distinct from non-invertible mimetic symmetry. We prove that a same-field sector $K(\phi,\widetilde X)$ preserves the local shift exactly when its scalar density is variationally trivial. For one scalar on a generic branch this requires constant $K$; nonconstant $K(\widetilde X)$ leaves only a global shift. The multi-field extension has local group $\mathbb R^n$ and admits determinant-type topological exceptions for $n\geq d$. Under standard local BV hypotheses, the spectator, its ghost and their antifields form a contractible quartet, so the spectator fibre adds no independent local counterterm, gauge deformation or anomaly class. The constant-map Einstein image is a special two-mode quartic-Horndeski subclass, while regular nonconstant $K$ restores one scalar. Finally, the complete canonical Einstein--scalar one-loop divergence shows that $(\widetilde\Box\phi)^2$ is equation-of-motion redundant at first loop order. The essential $\widetilde X^2$ pole pulls back to a first-derivative counterterm throughout the regular orbit and introduces no additional perturbative mode at $O(\hbar)$ within the stated hyperbolic EFT domain.