  This paper has two main goals. The first is universality of the KPZ equation
for fluctuations of dynamic interfaces associated to interacting particle
systems in the presence of open boundary. We consider generalizations on the
open-ASEP from [Corwin-Shen '16, Parekh '17] but admitting non-simple
interactions both at the boundary and within the bulk of the particle system.
These variations on open-ASEP are not integrable models, similar to the
long-range variations on ASEP considered in [Dembo-Tsai '15, Y '20]. We
establish the KPZ equation with the appropriate Robin boundary conditions as
scaling limits for height function fluctuations associated to these
non-integrable models, providing further evidence for the aforementioned
universality of the KPZ equation. We specialize to compact domains and address
non-compact domains in a second paper. The procedure that we employ to
establish the aforementioned theorem is the second main point of this paper.
Invariant measures in the presence of boundary interactions generally lack
reasonable descriptions. Thus, global analyses done through the invariant
measure, including the theory of energy solutions in [Goncalves-Jara '14,
Goncalves-Jara '17, Goncalves-Jara-Sethuraman '15], is immediately obstructed.
To circumvent this obstruction, we appeal to the almost entirely local nature
of the analysis in [Y '20].
