  We offer a new proposal for cosmic singularity resolution based upon a
quantum cosmology with a unitary bounce. This proposal is illustrated via a
novel quantization of a mini-superspace model in which there can be
superpositions of the cosmological constant. This possibility leads to a
finite, bouncing unitary cosmology. Whereas the usual Wheeler--DeWitt cosmology
generically displays pathological behaviour in terms of non-finite expectation
values and non-unitary dynamics, the finiteness and unitarity of our model are
formally guaranteed. For classically singular models with a massless scalar
field and cosmological constant, we show that well-behaved quantum observables
can be constructed and generic solutions to the universal Schr\"{o}dinger
equation are singularity-free. Generic solutions of our model displays novel
features including: i) superpositions of values of the cosmological constant;
ii) universal effective physics due to non-trivial self-adjoint extensions of
the Hamiltonian; and iii) bound `Efimov universe' states for negative
cosmological constant. The last feature provides a new platform for quantum
simulation of the early universe. A companion paper provides detailed
interpretation and analysis of particular cosmological solutions that display a
cosmic bounce due to quantum gravitational effects, a well-defined FLRW limit
far from the bounce, and a semi-classical turnaround point in the dynamics of
the scalar field which resembles an effective inflationary epoch.
