  The main objective of this paper is to propose a definition of non-connective
K-theory for a wide class of relative exact categories which, in general, do
not satisfy the factorization axiom and confirm that it agrees with the
non-connective K-theory for exact categories and complicial exact categories
with weak equivalences. The main application is to study the topological
filtrations of non-connective K-theory of a noetherian commutative ring with
unit in terms of Koszul cubes.
