A motivic analogue of the $K(1)$-local sphere spectrum

We identify the motivic KGL/2 -local sphere as the fiber of \psi^{3}-1 on (2,\eta) -completed Hermitian K -theory, over any base scheme containing 1/2 . This is a motivic analogue of the classical resolution of the K(1) -local sphere, and extends to a description of the KGL/2 -localization of an arbitrary motivic spectrum. Our proof relies on a novel conservativity argument that should be of broad utility in stable motivic homotopy theory.