Rigidity in etale motivic stable homotopy theory

For a scheme X, denote by SH(X-et boolean AND) the stabilization of the hypercompletion of its etale infinity-topos, and by SHet(X) the localization of the stable motivic homotopy category SH(X) at the (desuspensions of) etale hypercovers. For a stable infinity-category C, write C-p boolean AND for the p-completion of C. We prove that under suitable finiteness hypotheses, and assuming that p is invertible on X, the canonical functor e(p)boolean AND: SH(X-et boolean AND)(p)boolean AND -> SHet(X)(p)boolean AND is an equivalence of infinity-categories. This generalizes the rigidity theorems of Suslin and Voevodsky (Invent. Math. 123 (1996) 61-94), Ayoub (Ann. Sci. Ecole Norm. Sup. 47 (2014) 1-145) and Cisinski and Deglise (Compos. Math. 152 (2016) 556-666) to the setting of spectra. We deduce that under further regularity hypotheses on X, if S is the set of primes not invertible on X, then the endomorphisms of the S-local sphere in SHet(X) are given by etale hypercohomology with coefficients in the S-local classical sphere spectrum: [1[1/S], 1[1/S]](SHet(X)) similar or equal to H-et(0)(X, 1[1/S]). This confirms a conjecture of Morel. The primary novelty of our argument is that we use the pro-etale topology of Bhatt and Scholze (Asterisque 369 (2015) 99-201) to construct directly an invertible object (1) over cap (p)(1)[1] is an element of SH(X-et boolean AND)(p)boolean AND with the property that e(p)boolean AND((1) over cap (p)(1)[1]) similar or equal to Sigma(infinity)G(m) is an element of SHet(X)(p)boolean AND.