Nice triples and moving lemmas for motivic spaces

This paper contains geometric tools developed to solve the finite-field case of the Grothendieck–Serre conjecture in [1]. It turns out that the same machinery can be applied to solve some cohomological questions.
In particular, for any presheaf of $S^1$-spectra $E$ on the category of $k$-smooth schemes, all its Nisnevich sheaves of $\mathbf{A}^1$-stable homotopy groups are strictly homotopy invariant. This shows that $E$ is $\mathbf{A}^1$-local if and only if all its Nisnevich sheaves of ordinary stable homotopy groups are strictly homotopy invariant. The latter result was obtained by Morel [2] in the case when the field $k$ is infinite.
However, when $k$ is finite, Morel's proof does not work since it uses Gabber's presentation lemma and there is no published proof of that lemma. We do not use Gabber's presentation lemma. Instead, we develop the machinery of nice triples invented in [3]. This machinery is inspired by Voevodsky's technique of standard triples [4].