On Conformal Isometry of Grid Cells: Learning Distance-Preserving Position Embedding

L 1 has a non-zero minimum due to the non-infinitesimal range Conformal isometry is nearly exact for s∥∆x∥ ≤ 0.8 Beyond that point, deviation increases due to extrinsic curvature Minimizing L 1 finds the maximally distance-preserving embedding Key insight: The hexagon torus structure emerges as the optimal solution for preserving distances in all directions.