= Interval threshold strategy for discrete linear orders
{title2=$h_k=2^k-1$}
With $k$ moves remaining, preserve corresponding gap sizes exactly when smaller than $2^k-1$, and otherwise require both gaps to have at least that many points. One move splits a gap; $2^k-1=2(2^{k-1}-1)+1$ permits matching both smaller gaps. This strategy proves that $(\mathbb N,<)$ and $(\mathbb N+\mathbb Z,<)$ agree at every finite quantifier rank despite the second order containing points with infinitely many predecessors.
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