“equivalent” (DP) universality classes. Actually, it would be absurd to talk about the crossover between two models belonging to the identical universality class (see Sec. \[Sec:single\_to\_single\]). However, the DP${}_\textrm{IM}$ models differ from the DP models in regard to the “non-order” parameter at the transitions, which shows a discontinuous singularity at the crossover. This singularity induces a well-defined and nontrivial crossover from the DP${}_\textrm{IM}$ to the DP, more generally a crossover arising from a considerable reduction of the number of absorbing states (between two different DP${}_\textrm{IM}$).

We find two distinct crossover behaviors depending on the routes to reduce the number of absorbing states. Infinitesimal inclusion of an “excitatory” process (like single-particle branching) makes the system overly active, which gives rise to a discontinuous crossover (see Sec. \[Sec:dpima\]). While, the opposite “inhibitory” route (like single-particle annihilation) reveals a continuous crossover with the nontrivial crossover exponent $\phi\simeq 1.78(6)$. We find that this crossover exponent is universal for various kinds of models including the PCP and the triplet contact process (TCP). We argue that the crossover exponent is not independent but equal to $\nu_\|$, the relaxation time exponent of the DP universality class on a general footing. This conjecture is also confirmed in the 