  Learning compatible representations enables the interchangeable use of
semantic features as models are updated over time. This is particularly
relevant in search and retrieval systems where it is crucial to avoid
reprocessing of the gallery images with the updated model. While recent
research has shown promising empirical evidence, there is still a lack of
comprehensive theoretical understanding about learning compatible
representations. In this paper, we demonstrate that the stationary
representations learned by the $d$-Simplex fixed classifier optimally
approximate compatibility representation according to the two inequality
constraints of its formal definition. This not only establishes a solid
foundation for future works in this line of research but also presents
implications that can be exploited in practical learning scenarios. An
exemplary application is the now-standard practice of downloading and
fine-tuning new pre-trained models. Specifically, we show the strengths and
critical issues of stationary representations in the case in which a model
undergoing sequential fine-tuning is asynchronously replaced by downloading a
better-performing model pre-trained elsewhere. Such a representation enables
seamless delivery of retrieval service (i.e., no reprocessing of gallery
images) and offers improved performance without operational disruptions during
model replacement. Code available at: https://github.com/miccunifi/iamcl2r.
