Framed correspondences and the Milnor-Witt K-theory

The article is to construct a graded ring isomorphism between $H_0(ZF({\mathrm{\Delta }}_k^{\bullet },{\mathbb{G}}_m^{\wedge \ast }))$ and the Milnor-Witt K-theory ring $K_{\ast ⩾0}^{MW}(k)$, where $k$ is a field of characteristic zero and $Z{F}_{\ast }(k)$ is the category of linear framed correspondences of algebraic k-varieties, introduced by Garkusha and Panin. As it was shown by Garkusha and Panin, this partially recovers the computation of the motivic cohomotopy groups ${\pi }^{n,n}({\mathrm{\Sigma }}_{S^1}^{\mathrm{\infty }}{\mathrm{\Sigma }}_{{\mathbb{G}}_m}^{\mathrm{\infty }}{S}^0)(k)$, which is originally due to Morel.