I will describe an improved construction of the lattice regularization and cosmological reduction of the canonical formulation of loop quantum gravity. The application of this procedure to the Hamiltonian constraint provides its lattice analog, the domain of which has a natural structure of a sum of elementary cells. As a result, the related scalar constraint operator, which spectrum is independent of intertwiners, can be defined. The cosmological phase space reduction of lattice gravity requires a rigorous application of gauge-fixing conditions. The obtained Hamiltonian constraint is finite (without any cut-off introduction) and exact (with the holonomy expansion around the unit element of SU(2), known from QFT). It describes a simple structure of inhomogeneities and anisotropies. Consequently, the construction of the quantum evolution of the Universe in terms of transition amplitudes (instead of using perturbative approximations) appears to be possible.