Künneth formulas for motives and additivity of traces

We prove several K\"unneth formulas in motivic homotopy categories and deduce a Verdier pairing these following SGA5, which leads to the characteristic class of constructible motive, an invariant closely related Euler-Poincar\'e characteristic. additivity property using language derivators, approach May Groth-Ponto-Shulman; such result we show that presence Chow weight structure, for all motives is uniquely characterized by proper covariance, along distinguished triangles, refined Gysin morphisms Euler classes. In relative setting, under some transversality conditions, define class.