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At each step offer a fixed number of independent uniform candidate edges on a fixed vertex set, and let a rule select one. The standard two-choice model offers two candidates. The rule may depend on the entire previously exposed history. Repeated already present edges make no change in the independent-candidate version; another usual version offers only absent edges. The forced merging of large components and persistence of unsampled graph components apply to both over a linear number of steps. Fixed choice is essential in the continuity of fixed-choice percolation argument.

The persistence of unsampled graph components prevents the mass from each size band from disappearing entirely before $t_c$. More precisely, a band $[k,Dk)$ containing at least $\delta n/4$ vertices retains at least $\beta n$ vertices in that same band after at most $n/k$ steps, with high probability, for a fixed $\beta >0$ depending only on $\delta ,D$. To see why, each initial graph component in that band has a fixed positive probability that none of the offered edge endpoints touches it during the interval. Such a graph component remains unchanged regardless of the selection rule. The McDiarmid inequality concentrates the total mass of these untouched graph components. The reusable lemma supplies the full uniform-in-starting-time argument, including the version that samples absent edges.

Choose finitely many disjoint bands $[D^j,D^{j+1})$, $j=0,\dots ,J$, with $(J+1)\beta >1$. Part (i) provides at least $\delta n/4$ vertices in band $j$ at its time $t_{D^j}$. Each interval from that time to $t_c$ has length at most $n/D^j$, so persistence leaves at least $\beta n$ vertices in every one of these disjoint bands at the common time $t_c$. A union bound over this fixed finite number of bands makes all the conclusions simultaneous. Their total exceeds $n$, a contradiction. No fixed two-choice Achlioptas rule satisfies the explosive percolation hypothesis.

This continuity of fixed-choice percolation is the fixed-choice obstruction proved by Riordan and Warnke; their original research paper also treats a broader class of rules. The argument here does not apply when the number of offered choices grows with $n$.