Independent oriented-percolation comparison for the contact process ($q=e^{-2\delta }(1-e^{-\lambda \delta }{)}^2$)

For the one-dimensional contact process with recovery rate one and per-neighbour infection rate $\lambda$, use parity sites $(x,n)$ with $x+n$ even. Call a site good if $x$ has no recovery in $[(n-1)\delta ,(n+1)\delta ]$ and sends an arrow to each neighbour during $[n\delta ,(n+1)\delta ]$. The goodness indicators are independent: recovery intervals for successive parity sites at the same spatial point only touch at endpoints, and all outgoing-arrow intervals and clocks are disjoint. Their common probability is the displayed $q$. A path of good sites carries infection from each time layer to the next, because both its source and target avoid recovery during the transmitting interval. Taking $\delta ={\lambda }^{-1/2}$ makes $q\to 1$. The incoming-edge coupling of site and bond percolation and $p_c^{\mathrm{bond}}<1$ imply $p_c^{\mathrm{site}}<1$, so this proves a finite critical infection rate.