Neurons in the primary motor cortex (M1) respond to short hand trajectories — fragments of movement, not individual joint angles or muscle activations. These fragments have both geometric properties (curvature, direction) and kinematic properties (speed, acceleration). Experimental data shows that geometric and kinematic features are coupled: speed systematically varies with curvature in a way known as the two-thirds power law. But why the motor cortex encodes this coupling, rather than representing geometry and kinematics independently, has lacked a mathematical explanation.
Sub-Riemannian geometry provides one (arXiv:2603.20756). In a sub-Riemannian space, movement is constrained to certain directions at each point — not all motions are allowed. The researchers construct a higher-dimensional geometry where the constraints encode both position and velocity information simultaneously. Horizontal curves in this space — paths that satisfy the constraints — automatically satisfy the observed coupling between geometric and kinematic properties. The two-thirds power law isn't imposed; it emerges from the geometry.
The choice of distance metric matters. When clustering trajectories in this geometric space, the Wasserstein distance (optimal transport) groups movement fragments into categories that match experimental neural data far more efficiently than the Sobolev distance (function space norm). The Wasserstein distance compares distributions of trajectory mass; the Sobolev distance compares point-by-point function values. The cortex appears to care about the transport structure, not the pointwise structure.
The structural insight: the motor cortex's encoding of movement fragments is not a neural code to be deciphered but a geometry to be recognized. The constraints of the sub-Riemannian space generate the observed statistics as necessary consequences, not contingent implementations. The coupling between shape and speed that the cortex preserves is the coupling that the geometry demands. The brain didn't learn this relationship — it instantiates a space where the relationship is axiomatic.