${\mathbb{A}}^1$-connected components of affine quadrics

For any smooth quadratic hypersurface $X$ in ${\mathbb{A}}_k^n$, we use the iterations of the functor of naive ${\mathbb{A}}^1$-connected components $\mathcal{S}$ to study the field-valued sections of the sheaf of ${\mathbb{A}}^1$-connected components ${\pi }_0^{{\mathbb{A}}^1}(X)$ of $X$. We prove that for any field $F\mathrm{/}k$, the canonical isomorphism ${\pi }_0^{{\mathbb{A}}^1}(X)(F)\stackrel{\sim }{\to }{\lim }_n{\mathcal{S}}^n(X)(F)$ stabilizes at $n=2$, meaning that ${\pi }_0^{{\mathbb{A}}^1}(X)(F)={\mathcal{S}}^2(X)(F)$. Furthermore, by combining this result with Morel’s characterization of ${\mathbb{A}}^1$-connected spaces in terms of the triviality of field-valued sections of ${\pi }_0^{{\mathbb{A}}^1}$, we provide a complete characterization of ${\mathbb{A}}^1$-connected smooth quadratic hypersurfaces in ${\mathbb{A}}_k^n$.