Our research concerns the development of a theoretical framework bridging Finite Element Exterior Calculus (FEEC) (citation) with Cut Finite Element Methods (CutFEM) (citation) using a fully discrete approach (citation). The work introduces theoretical tools and applies them to study a discrete unfitted method for the Hodge Laplace equation. Our approach helps to synthesize and generalize currently available theoretical error analysis for CutFEM. Another benefit is the ability to treat domains with nontrivial topology in a way which respects the continuous homological structure. The latter is a feature which has not been investigated thoroughly in the CutFEM literature hitherto this work. More specifically, the fully discrete approach affords us simpler error analysis, and the use of differential forms allows to generalize the mixed stabilization term introduced in (citations) which among other things, in the case of $k=n-1$, where $n$ is the dimension of the domain, allows to resolve a divergence condition to pointwise machine precision.