Connectivity and Purity for logarithmic motives

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 ${\mathbf{l}\mathbf{o}\mathbf{g}\mathbf{D}\mathbf{M}}^{\text{eff}}(k)$ is proved to be compatible with Voevodsky’s $t$-structure i.e. we show that the comparison functor $R^{\overline{\square}}\omega^*\colon $\mathbf{D}\mathbf{M}$^{$\text{eff}$}(k)\to $\mathbf{l}\mathbf{o}\mathbf{g}\mathbf{D}\mathbf{M}$^{$\text{eff}$}(k)$is$t$-exact. The heart of the homotopy $t$-structure on ${\mathbf{l}\mathbf{o}\mathbf{g}\mathbf{D}\mathbf{M}}^{\text{eff}}(k)$ 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.