Classification of generalized quasitopological gravities

Generalized quasitopological gravities (GQTGs) are higher-order extensions of Einstein gravity in $D$ dimensions satisfying a number interesting properties, such as possessing second-order linearized equations motion on top maximally symmetric backgrounds, admitting nonhairy generalizations the Schwarzschild-Tangherlini black hole which characterized by single metric function or forming perturbative spanning set space effective theories gravity. In this work, we classify all inequivalent GQTGs at curvature orders $n$ and spacetime dimension $D\ensuremath{\ge}4$. This is achieved after explicit construction dictionary that allows uplift expressions evaluated single-function static spherically ansatz into fully covariant ones. On one hand, applying prescription for $D\ensuremath{\ge}5$, find form unique exists each and, first time, $n\ensuremath{-}2$ proper existing every order $n$. other $D=4$, able to provide rigorous proof fact there one, only (proper) GQTG $n$, deriving along way simple expression four-dimensional representative