$\mathbb{A}^1$-connectivity of motivic spaces

We prove a version of Morel's unstable $\mathbb{A}^1$-connectivity theorem over arbitrary base schemes.

In the stable setting, this recovers (and simplifies the proof of) the known connectivity bounds due to Morel, Schmidt--Strunk, Deshmukh--Hogadi--Kulkarni--Yadav, and Druzhinin, and extends them to possibly non-noetherian schemes.

Using the recent work of Bachmann--Elmanto--Morrow, this also implies that the slice filtration on homotopy $K$-theory is convergent for qcqs schemes of finite valuative dimension.