A hard limit on geometrizing quantum cosmology models

The Eisenhart-Duval lift recasts classical dynamics as geodesic motion in a higher-dimensional spacetime — but only when the Hamiltonian is quadratic in momenta. Polymer-modified Hamiltonians from Loop Quantum Cosmology violate that condition, becoming non-polynomial and thus ineligible for this geometric treatment. The result is a kinematical no-go theorem: any cosmological model with quantum corrections of this type sits outside the reach of Eisenhart-Duval geometrization entirely. The finding draws a clear boundary around a widely used technique, signaling that quantum-corrected bounce scenarios require different geometric tools.