Hypercomplete étale framed motives and comparison of stable homotopy groups of motivic spectra and étale realizations over a field

For any base field and integer $l$ invertible in $k$, we prove that ${\mathrm{\Omega }}_{{\mathbb{G}}_m}^{\mathrm{\infty }}$ and ${\mathrm{\Omega }}_{{\mathbb{P}}^1}^{\mathrm{\infty }}$ commute with hyper étale sheafification $L_{\acute{e}t}$ and Betti realization through infinite loop space theory in motivic homotopy theory. The central subject of this article is an $l$-complete hypercomplete étale analog of the framed motives theory developed by Garkusha and Panin. Using Bachman’s hypercomplete étale \RigidityTheorem and the $\mathrm{\infty }$-categorical approach of framed motivic spaces by Elmanto, Hoyois, Khan, Sosnilo, Yakerson, we prove the recognition principle and the framed motives formula for the composite functor $${\mathrm{\Delta }}^{\mathrm{o}\mathrm{p}}{\mathrm{S}\mathrm{m}}_k\to {\mathrm{S}\mathrm{p}\mathrm{t}}_{{\mathbb{A}}^1,\acute{e}t}^{{\mathbb{G}}_m^{-1}}({\mathrm{S}\mathrm{m}}_k)\stackrel{{\mathrm{\Omega }}_{{\mathbb{G}}_m}^{\mathrm{\infty }}}{\to }{\mathrm{S}\mathrm{p}\mathrm{t}}_{\acute{e}t,\widehat{n}}({\mathrm{S}\mathrm{m}}_k)\mathrm{.}$$ The first applications include the hypercomplete étale stable motivic connectivity theorem and an étale local isomorphism $${\pi }_{i,j}^{{\mathbb{A}}^1,\mathrm{N}\mathrm{i}\mathrm{s}}(E)\simeq {\pi }_{i,j}^{{\mathbb{A}}^1,\acute{e}t}(E)$$ for any $l$-complete effective motivic spectra $E$, and $j\ge 0$. Furthermore, we obtain a new proof for Levine’s comparison isomorphism over $\mathbb{C}$, ${\pi }_{i,0}^{{\mathbb{A}}^1,\mathrm{N}\mathrm{i}\mathrm{s}}(E)(\mathbb{C})\cong {\pi }_i(Be(E))$, and Zargar’s generalization for algebraically closed fields, that applies to an arbitrary base field.