= Solution
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 $[D^j,D^{j+1})$, $j=0,\ldots,J$, 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. \b[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 https://arxiv.org/abs/1102.5306[original research paper] also treats a broader class of rules. The argument here does not apply when the number of offered choices grows with $n$.
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