Is the brain a 1-dimensional diffeological space?

A new mathematical paper by Patrick Iglesias-Zemmour, titled 'Is the brain a 1-dimensional diffeological space?', proposes a radical shift in how we model the brain's geometry for motor control. For decades, neuroscience has observed that human hand movements obey the Two-Thirds Power Law, where velocity is proportional to curvature raised to the -1/3 power. Geometrically, this implies trajectories are geodesics of an 'equi-affine' metric. However, this metric has been mathematically problematic because its definition relies on acceleration, which isn't a proper tensor under standard geometry. Iglesias-Zemmour's work, submitted to Biological Cybernetics, applies the framework of 'Diffeology'—specifically a construct called the 'Wire Plane'—to resolve this inconsistency.

The key innovation is modeling the brain's internal geometry not as the standard 2D plane (R²), but as the Wire Plane: R² equipped with a 'wire diffeology' generated by smooth curves. In this framework, the problematic equi-affine metric is founded upon a covariant 3-tensor that becomes a natural, intrinsic object. This leads to the neuro-geometric hypothesis that the brain perceives and plans spatial movement through 1-dimensional paths (like wires), rather than assembling 2-dimensional coordinate charts. The 7-page proof suggests our neural representation of space for action might be fundamentally path-based, which could influence future models in robotics, AI motion planning, and our understanding of neurological disorders affecting movement.