= Handle-slide isolation of a cancelling pair
{title2=$\begin{pmatrix}\delta&0\\0&D-cb/\delta\end{pmatrix}$}
In the intermediate-index range, if an <attaching sphere> $A_1$ meets a <belt sphere> $B_1$ at exactly one transverse point, <handle slides> can remove the intersections of every other attaching sphere with $B_1$. The dual slides then remove the intersections of $A_1$ with the other belt spheres. Each move uses a band to the unique pivot sheet and a local isotopy removing the old and new intersections. If $m_{ji}=A_j\cdot B_i$ and $\delta=m_{11}=\pm1$, the remaining incidence block is $m_{ji}-m_{j1}m_{1i}/\delta$, the <Schur complement> of the pivot. Extreme-index cases need separate hypotheses.
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