Hypercomplete \'etale framed motives and comparison of stable homotopy groups of motivic spectra and \'etale realizations over a field

For any base field and integer l invertible in k, we prove that ^₆䂷 and ^₏℉ commute with hyper \'etale sheafification L₄ₓ and Betti realization through infinite loop space theory in motivic homotopy theory. The central subject of this article is an l-complete hypercomplete \'etale analog of the framed motives theory developed by Garkusha and Panin. Using Bachman's hypercomplete \'etale and the -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 \ᵒpSmₖ Spt^Gₘ^{-1}₀℉, ₄ₓ (Smₖ) ^₆䂷 Spt₄ₓ, ₍ (Smₖ). \ The first applications include the hypercomplete \'etale stable motivic connectivity theorem and an \'etale local isomorphism \^A¹, Nis₈, ₉ (E) ^A¹, et₈, ₉ (E) \ for any l-complete effective motivic spectra E, and j 0. Furthermore, we obtain a new proof for Levine's comparison isomorphism over C, ₈, ₀^A¹, Nis (E) (C) ᵢ (Be (E) ), and Zargar's generalization for algebraically closed fields, that applies to an arbitrary base field.