  The goal of this paper is to extend the work of Voevodsky and Morel on the
homotopy $t$-structure on the category of motivic complexes to the context of
motives for logarithmic schemes. To do so, we prove an analogue of Morel's
connectivity theorem and show a purity statement for $(\mathbf{P}^1,
\infty)$-local complexes of sheaves with log transfers.
  The homotopy $t$-structure on $\mathbf{logDM}^{\textrm{eff}}(k)$ is proved to
be compatible with Voevodsky's $t$-structure i.e. we show that the comparison
functor $R^{\overline{\square}}\omega^*\colon \mathbf{DM}^{\textrm{eff}}(k)\to
\mathbf{logDM}^{\textrm{eff}}(k)$ is $t$-exact.
  The heart of the homotopy $t$-structure on $\mathbf{logDM}^{\textrm{eff}}(k)$
is the Grothendieck abelian category of strictly cube-invariant sheaves with
log transfers: we use it to build a new version of the category of reciprocity
sheaves in the style of Kahn--Saito--Yamazaki and R\"ulling.
