Foundation machine learning interatomic potentials (MLIPs) are trained on overlapping chemical spaces, yet their latent representations remain model-specific. Here, we show that independently developed MLIPs exhibit statistically consistent geometric organisation of atomic environments, which we term the Platonic representation. By projecting embeddings relative to a set of atomic anchors, we unify the latent spaces of seven MLIPs (spanning equivariant, non-equivariant, conservative, and non-conservative architectures) into a common metric space that preserves chemical periodicity and structural invariants. This unified framework enables direct cross-model optimal transport, interpretable embedding arithmetic, and the detection of representational biases. Furthermore, we demonstrate that geometric distortions in this space can indicate physical prediction failures, including symmetry breaking and incorrect phonon dispersions. Our results show that the latent spaces of diverse MLIPs present consistent statistical geometry shaped by shared physical and chemical constraints, suggesting that the Platonic representation offers a practical route toward interoperable, comparable, and interpretable foundation models for materials science.

A connectional brain template (CBT) is a holistic representation of a population of multi-view brain connectivity graphs, encoding shared patterns and normalizing typical variations across individuals. The federation of CBT learning allows for an inclusive estimation of the representative center of multi-domain brain connectivity datasets in a fully data-preserving manner. However, existing methods overlook the non-independent and identically distributed (non-IDD) issue stemming from multidomain brain connectivity heterogeneity, in which data domains are drawn from different hospitals and imaging modalities. To overcome this limitation, we unprecedentedly propose a metadata-driven federated learning framework, called MetaFedCBT, for cross-domain CBT learning. Given the data drawn from a specific domain (i.e., hospital), our model aims to learn metadata in a fully supervised manner by introducing a local client-based regressor network. The generated meta-data is forced to meet the statistical attributes (e.g., mean) of other domains, while preserving their privacy. Our supervised meta-data generation approach boosts the unsupervised learning of a more centered, representative, and holistic CBT of a particular brain state across diverse domains. As the federated learning progresses over multiple rounds, the learned metadata and associated generated connectivities are continuously updated to better approximate the target domain information. MetaFedCBT overcomes the non-IID issue of existing methods by generating informative brain connectivities for privacy-preserving holistic CBT learning with guidance using metadata. Extensive experiments on multi-view morphological brain networks of normal and patient subjects demonstrate that our MetaFedCBT is a superior federated CBT learning model and significantly advances the state-of-the-art performance.