Conformal normalization provably yields a torus, only favoring hexagonal grids

Xu et al. train path-integration RNNs with conformal normalization, rescaling input velocity so local neural-state displacement is direction-independently proportional to physical displacement [xu-etal-2024-conformal-grid-cells] They prove the transformations form a representation of the abelian Lie group (R^2,+), and since firing rates are bounded the population manifold is a compact connected abelian group, hence topologically a torus [xu-etal-2024-conformal-grid-cells] Hexagonality is only favored, not proven: a Fourier packing argument shows the hexagonal lattice fits the kernel better than the square, and hexagonal grids emerge numerically (gridness 0.86-0.87) [xu-etal-2024-conformal-grid-cells] Conformal normalization is a train-time constraint the geometry depends on: a train-time ablation (training without it) yields non-hexagon or stripe-like patterns instead, so the torus/hexagon structure is imposed rather than freely emergent [xu-etal-2024-conformal-grid-cells]