  Morel's stable connectivity theorems state that for any connective
$S^1$-spectrum $F$ of motivic spaces (Nisnevich simplicial sheaves) over an
arbitrary field, the spectrum $L_{\mathbb A^1}(F)$ is connective, and the same
property for $\mathbb P^1$-spectra of motivic spaces. Here $L_{\mathbb A^1}$
denotes the $\mathbb A^1$-localisation in the category of motivic spectra over
a field $k$.
  Originally the same property was conjectured for the case of motivic
$S^1$-spectra over a base scheme $S$.In view of Ayoub's conterexamples the
modified version of conjecture states that $L_{\mathbb A^1}(F)$ is
$(-d)$-connective for any connective $F$, where $d=\mathrm{dim} S$ is the Krull
dimension. The conjecture is proven under the infiniteness assumption on the
residue fields for the cases of Dedekind schemes by J.~Schmidt and F.~Strunk
and noetherian domains of arbitrary dimension by N.~Deshmukh, A.~Hogadi,
G.~Kulkarni and S.~Yadavand. In the article we prove the result or general base
with out the assumption on the residue fields.
  So by the result for any smooth scheme $X$ over a base scheme $S$ of Krull
dimension $d$ the Nisnevich sheaves of $S^1$-stable motivic homotopy groups
$\pi_i^{S^1}(X)$ and $\mathbb P^1$-stable motivic homotopy groups
$\pi_{i+j,j}^{\mathbb P^1}(X)$ vanishes for all $i<-d$.
