  We set up an Einstein-Gauss-Bonnet theory in four dimensions, based on the
recent formulation of pure gravity with extra dimensions of vanishing metrical
length [1]. In absence of torsion, the effective field equations depend only on
the four-metric, reflecting a quadratic curvature nonlinearity but no higher
than second order derivatives. In contrast with recent proposals to obtain
4dEGB theories through a singular rescaling of the Gauss-Bonnet coupling, this
formalism requires no (classical) regularization of divergences and is
inequivalent to Horndeski gravity, while being generally covariant and
independent of compactification. Notably, the vacuum field equations admit FLRW
cosmologies containing nonsingular bounce and self-accelerating Universe, and
spherically symmetric black holes more general than Schwarzschild. The
quadratic theory of gravity emerging here is shown to be preserved against the
inclusion of higher Lovelock densities in 5+2n dimensions.
