We prove the first unconditional amortized $Ω\big((\log N/\log\log N)^2\big)$ cell-probe lower bound for a Boolean dynamic problem. For insert-only multiphase Disjointness with $N$ query sets over $[N]$ and word size $w=Θ(\log N)$, a fixed distribution of interleaved insertions and queries forces this expected average cost per operation, excluding preprocessing. The bound allows sufficiently small constant error for each query of every fixed legal sequence and imposes no restriction on individual operation costs. It transfers to incremental directed reachability, incremental subgraph connectivity, and fully dynamic triangle detection. We also prove $t_q=Ω\big((\log N/\log(t_uw))^2\big)$ for amortized insertion time $1\le t_u=N^{o(1)}$ and expected query time $t_q$, when the number of insertions before the final query is not announced in advance. We reach the quadratic-logarithmic scale with bounded error by combining interleaving with the 2.5-round communication framework [Ko26a, Ko26b] and controlling errors among accepted simulations.

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