Motivic infinite loop spaces

We prove a recognition principle for motivic infinite P1-loop spaces over perfect field. This is achieved by developing theory of framed spaces, which analogue the E-infinity-spaces. A space equipped with transfers along finite syntomic morphisms trivialized cotangent complex in K-theory. Our main result that grouplike are equivalent to full subcategory spectra generated under colimits suspension spectra. As consequence, we deduce some representability results smooth varieties, and particular sphere spectrum, terms Hilbert schemes points affine spaces.