𝔸1-Connected Components of Affine Quadrics

For any smooth quadratic hypersurface X in Aⁿₖ, we use the iterations of the functor of naive A¹-connected components S to study the field-valued sections of the sheaf of A¹-connected components π ₀^A¹(X) of X. We prove that for any field $F/k$, the canonical isomorphism π ₀^A¹(X)(F) ~ lim ₙ Sⁿ(X)(F) stabilizes at $n=2$, meaning that π ₀^A¹(X)(F)=S²(X)(F). Furthermore, by combining this result with Morel’s characterization of A¹-connected spaces in terms of the triviality of field-valued sections of π ₀^A¹, we provide a complete characterization of A¹-connected smooth quadratic hypersurfaces in Aⁿₖ.