Record-level versus client-level adjacency

List the client’s 100 occurrences as $u_1,\dots ,u_{100}$, with repetitions allowed. For $j=0,\dots ,100$, let $D_j=D_0+{\sum }_{\mathrm{\ell }=1}^j{\mathbf{1}}_{\{u_{\mathrm{\ell }}\}}$. Then $D_{j-1}$ and $D_j$ differ by adding one occurrence, so [F1] makes each consecutive pair neighbors.

The count-function difference between $D_0$ and $D_{100}$ has total absolute count ${\sum }_{u\in U}\mathrm{\mid }{D}_{100}(u)-D_0(u)\mathrm{\mid }=100$, counting repeated contributions with multiplicity. A single add/remove neighbor pair has this total equal to $1$ by [F1]. Hence the full contribution is not a record-level neighboring step. Treating it as one step changes the adjacency relation, and [F2] says that any claimed client-level guarantee needs that relation and a proof under it.