Alignment-Free Geometry Of Representational Trajectories Across The Brain And Foundation Models

We introduce an alignment-free Riemannian framework for comparing representational trajectories across systems that share neither coordinates, dimensionality, nor a common axis of evolution. We validate it by analyzing the response of the visual cortex and foundation models on common images. We study EEG and MEG responses to natural images from THINGS-EEG2 and THINGS-MEG, together with a label-fed language model, Llama-3-8B, and an image-fed vision model, DINOv2. For each system, an image is represented by a trajectory of neighborhood covariances on the symmetric positive-definite manifold. Each system's covariance trajectories are mapped to the tangent space at that system's Fr\'echet mean, velocity directions are computed along the system-specific axis, and systems are compared only through the pairwise angular geometry of per-image velocity directions. This statistic is invariant to an independent invertible congruence of each system's PCA-reduced covariances and to per-image rescaling. Across both EEG and MEG, cortical trajectories share positive angular geometry with both Llama and DINOv2. Order-sensitive tests separate two kinds of correspondence: the best-matching depth does not advance steadily in either model, while a rate-free monotone test finds cortical time and model depth ordered against the label-fed model in both cohorts and not against the image-fed one. Category analysis shows the correspondence is structured by graded neighborhood relations rather than reducible to discrete category labels, while scaffold controls reveal a substantial fraction of slice-wise concordance is induced by the shared stimulus-neighborhood graph. Methodologically, the framework separates coordinate-free geometric correspondence from order-preserving dynamical correspondence, providing a general tool to test when two representational systems share structure and trajectory.