A 1 A¹double struck upper A Superscript 1-CONNECTIVITY OF MOTIVIC SPACES

We prove a version of Morel’s unstable A 1 A¹ double struck upper A Superscript 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.