How Hyperbolic Are Hyperbolic Vision-Language Models?

Hyperbolic vision-language models (VLMs) have emerged as a rapidly growing area of research, reporting strong zero-shot performance alongside an interpretable embedding space in which radius orders representations by generality. But how hyperbolic are hyperbolic VLMs in practice? We investigate this at the exponential map, the single operation through which curvature enters an otherwise Euclidean network. We show that across MERU, HyCoCLIP, ARGENT, and PHyCLIP, the exponential map is well approximated by the identity function, i.e., pairwise distances and angles in the pre-map Euclidean space barely change under projection, and removing the map at inference leaves evaluation scores unchanged. Motivated by this observation, we train Euclidean VLMs on CC3M from scratch under the same hierarchy losses. Over roughly the final $90$\% of the training steps, the training statistics are nearly indistinguishable from those of their hyperbolic counterparts, and the resulting models match the evaluation scores and recover the same radial distributions. Our findings establish Euclidean space as a sufficient substrate for hierarchical vision-language learning as it is currently practiced, and motivate new paradigms in which hyperbolic geometry can deliver on its theoretical promise. Code is available in our \href{https://anonymous.4open.science/r/hyperbolic-vlms-4780/README.md}{GitHub}.