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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.

Independently open the bonds of a directed graph with a common probability $p$, and ask whether an infinite open directed path starts at a specified root. On the oriented square lattice, edges run from $(x,n)$ to $(x\pm 1,n+1)$ with $x+n$ even. The critical probability is the infimum of parameters with positive survival probability. The incoming-edge coupling of site and bond percolation compares this threshold with that of oriented site percolation.

View the oriented square lattice as $(x,n)$ with $x+n$ even, and arrows $(x,n)\to (x\pm 1,n+1)$. Each graph vertex has two incoming bonds. Start with independent oriented bond percolation of parameter $p$, and declare a site open when at least one of its incoming bonds is open. Its probability of being open isIncoming bond sets for different heads are disjoint, so these site indicators are independent: they are precisely an oriented site percolation configuration with parameter $r$.

Every graph vertex after the first on an open oriented bond path is open as a site, because its incoming path bond is open. The root's two incoming bonds belong to earlier time layers and are independent of all bonds used by a forward path from the root. Thus the event that the root site is open, of probability $r$, is independent of bond survival from that root. On their intersection the entire infinite bond path is also an infinite open site path. Hence, writing ${\theta }_b,{\theta }_s$ for the two survival probabilities,This is the incoming-edge coupling of site and bond percolation. If site survival is defined conditional on an open root, the extra factor is omitted; the critical probability is the same. If the model is drawn only in a time half-plane, one can independently sample the root's incoming bonds as auxiliary variables.

For every $p>p_c(\mathrm{bond})$, bond survival is positive by monotonicity and the definition of the critical probability. Therefore $p_c(\mathrm{site})\le 1-(1-p{)}^2$. Let $p$ decrease to the bond critical probability and use continuity of this polynomial, obtainingIf the bond critical probability equals one, the inequality is trivial; no assertion about survival exactly at criticality is used.

Use the incoming-edge coupling of site and bond percolation. Start with independent random variables indicating open bonds, each with probability $q$. For a vertex $v$ of incoming degree $d_v$, let $J_v$ indicate that at least one incoming bond is open. ThenThe indicators $J_v$ are independent: the sets of bonds entering different vertices are disjoint. Using an additional independent coin at each vertex, turn some zeros into ones so that the resulting indicators $S_v$ are independent with common probability $r$. Explicitly, when $J_v=0$, set $S_v=1$ with conditional probability $(r-\mathbb{P}(J_v=1))/(1-\mathbb{P}(J_v=1))$.

Every vertex other than $x$ in an open bond percolation out-cluster has an open incoming bond, and is consequently an open site. On $\{S_x=1\}$, the whole bond out-cluster is therefore contained in the site percolation out-cluster. The event of infinite bond reachability does not depend on bonds entering $x$: delete cycles from any finite open directed path starting at $x$. It is thus independent of $S_x$, including its additional coin. HenceFor every $q>p_H^b$ below $1$, monotonicity and the definition of the survival threshold give ${\theta }_b(q)>0$. Letting $q\downarrow p_H^b$ yields the quantitative boundIn particular, if $p_H^b<1$, the right-hand side is strictly less than $1$. If $p_H^b=1$, no separation from $1$ is implied or generally possible.