Same World, Different Geometry: A Controlled Test of the Platonic Representation Hypothesis

The Platonic Representation Hypothesis Platonic holds that models trained on different modalities converge, with scale and competence, toward a common representation of the world. Testing whether models converge to the world, and not only to each other, requires knowing the world's true structure, which is unknown for natural images and text. Consequently, existing evidence compares representations only with each other, not with the world, and cannot manipulate what makes them agree. We therefore construct a world whose structure is known: eight Hamiltonian systems with ground-truth latent variables $h$, rendered independently as video and text. The two encoders share no weights or joint objective and are trained in matched pairs that differ only in their prediction targets, so the cause of alignment can be manipulated directly. We show that a shared world is not enough: each encoder recovers the physics, yet the two do not align. Alignment arises only when both are also trained to predict information about $h$. We characterize this phenomenon through the $H$-subspace, the subspace of each latent on which both encoders carry the same linear statistics of $h$. We prove that the latent spaces align on it and show that a single linear map transports information between the spaces. Alignment depends on the information the targets carry about $h$, not on whether they are shared: different targets, each informative about $h$, align the encoders, while an identical target uninformative about $h$ does not. Large-scale pretraining does not substitute for this information: models pretrained on unrelated natural corpora decode the physics but share no such subspace. In this setting, convergence is installed by information about $h$, not by observing the same world.