\end{aligned}$$ The above model has infinitely many absorbing states, but with nonzero $w$ the number of absorbing states is greatly reduced. At $w=0$, there is again a jump in the auxiliary field density (here, the pair density). We found that the critical point for the TCP at $w=0$ is $p_c = 0.036~865(5)$, exploiting the DP critical scaling (data not shown). Figure \[Fig:TCPcollapse\] shows the scaling plot of the triplet density $\rho_t$ in the same way as in Fig. \[Fig:mucollapse\]. We also measured the crossover exponent from the phase boundary and found the same exponent (data not shown).

Hence we conclude that there is the well-defined and universal crossover scaling from the DP${}_\textrm{IM}$ to the DP which is mediated by the significant reduction of the number of absorbing states. The discontinuity in the auxiliary field plays a crucial role in this nontrivial crossover.

\[Sec:discussion\] Conjecture on the crossover exponent $\phi$
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The crossover exponent from the DP${}_\textrm{IM}$ to the DP is estimated as $\phi = 1.78(6)$. Since this crossover occurs between the same universality class, we are suspicious that $\phi$ may not be independent but related to the well-known DP critical exponents. Actually, we argue that the crossover exponent is given 