Scale-invariant Schrödinger geometry in symmetric teleparallel gravity

We construct a locally scale-invariant formulation of Schr\"{o}dinger geometry in symmetric teleparallel gravity. The unique scale transformation preserving autoparallelism, torsionlessness and the Schr\"{o}dinger affine structure is first identified, from which a quadratic scale-invariant action is obtained. Using the Palatini formalism, we show that the conditions required for scale invariance are exactly those ensuring that the affine connection is dynamically reduced to the Schr\"{o}dinger form. Our results establish a direct correspondence between local scale symmetry and length-preserving affine geometry, providing a new geometric framework for scale-invariant metric-affine gravity.