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 ${\tilde{g}}_{\mu \nu }=C(\varphi ,X)g_{\mu \nu }-D(\varphi ,X){\nabla }_{\mu }\varphi {\nabla }_{\nu }\varphi$, which pulls the spectator translation back to an exact local gauge redundancy. We derive its closed generator and show that $\mathcal{J}=C-X{C}_X+X^2{D}_X$ controls kinetic invertibility, the generator denominator and the metric-map determinant. This regular ($\mathcal{J}\ne 0$) construction is distinct from non-invertible mimetic symmetry. We prove that a same-field sector $K(\varphi ,\tilde{X})$ preserves the local shift exactly when its scalar density is variationally trivial. For one scalar on a generic branch this requires constant $K$; nonconstant $K(\tilde{X})$ leaves only a global shift. The multi-field extension has local group ${\mathbb{R}}^n$ and admits determinant-type topological exceptions for $n\ge d$. 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 $(\tilde{\square }\varphi {)}^2$ is equation-of-motion redundant at first loop order. The essential ${\tilde{X}}^2$ pole pulls back to a first-derivative counterterm throughout the regular orbit and introduces no additional perturbative mode at $O(\hslash )$ within the stated hyperbolic EFT domain.

results significantly strengthen, symmetry is now a theorem; findings expanded to larger class of DHOST theories