Reflex instabilities I: Boundaries makes grid-based disc simulations frame-dependent

If the mass distribution within an accretion disc is asymmetric, it will exert a gravitational pull on its central object. The central object must then move away from the origin of the reference frame, resulting in atypical gravitational pulls on the disc. The goal of the present series of papers is to investigate the stability of this feedback loop. In this first paper, we report a surprising finding: the choice of reference frame affects the number and nature of the waves and instabilities contained in the analytical equations. To show this, we use a simple example: the modes discovered by Sanchez et al. (2025, Appendix A.3). We provide the first analytical description of those modes, we explain their physical mechanism, and propose to call them 'bracket modes' because they arise from the bracket side of an integration by parts. We show that the same disc can be stable in the stellocentric reference frame but unstable in the barycentric reference frame. This is because the boundaries of the simulation domain are frame-dependent. To avoid this, one must model the entire disc all the way from the central cavity to the outer edge. But this is prohibitively expensive, and some discs do not have a central cavity anyway. Our results suggest that extreme caution is required when interpreting simulations of asymmetric discs, and that our community may not have the tools to model the reflex motion of the central object yet.