 Dimension comparison for the contact process

Map a site $x\in {\mathbb{Z}}^d$ to its height $h(x)={\sum }_i{x}_i$. Maintain one infected representative in height $n$ for every infected site of an auxiliary one-dimensional contact process. The representative's death makes that auxiliary site recover. Its $d$ outgoing arrows to each neighbouring height, each of rate $\lambda$, make an auxiliary infection at total rate $d\lambda$, using the arrow's target as the new representative. Arrows to an already infected auxiliary height do nothing there. Distinct active representatives use distinct death and outgoing-arrow clocks, so their conditional rates are exactly those of the one-dimensional process. Every auxiliary infection has a live representative upstairs, proving the survival comparison.