Hofer nondegeneracy from rational Lagrangian displacement

Assume the lower bound $e(L)\ge \gamma (L)/2$ for a closed rational Lagrangian submanifold, with matching energy convention. A nonidentity Hamiltonian map displaces a sufficiently small ball about some moved point. Inside that ball place a Lagrangian torus formed from equal-area circles. Its least positive Liouville period is positive, so the map's norm is at least half this period. This proves positivity of the distance between different maps without appealing to nondegeneracy itself.