A local geodesic constant in a cayley graph

By F1, for every $s\le t$ in $I$ the path between them is the unique geodesic, of length $t-s$. Hence $d(q(s),q(t))=t-s$. The two $(1,0)$ inequalities are both this equality, and restricting to $t-s\le k$ proves locality for each $k$. In particular this supplies a concrete zero-slim instance of F2 for any positive radius, such as $k=1$.

For example, in the free group on $a,b$ the reduced path labelled $ab{a}^{-1}b$ has length $4$ and endpoint word length $4$. Its subpath between parameters $1\mathrm{/}2$ and $13\mathrm{/}4$ has distance $13\mathrm{/}4-1\mathrm{/}2=11\mathrm{/}4$, even though both endpoints lie inside edges. A one-letter path has endpoint distance $1$, and a zero-length restriction has distance $0$. These are the same equality from step 1.1, without additive error.