  An ideal invariant for multiparameter persistence would be discriminative,
computable and stable. In this work we analyse the discriminative power of a
stable, computable invariant of multiparameter persistence modules: the fibered
bar code. The fibered bar code is equivalent to the rank invariant and encodes
the bar codes of the 1-parameter submodules of a multiparameter module. This
invariant is well known to be globally incomplete. However in this work we show
that the fibered bar code is locally complete for finitely presented modules by
showing a local equivalence of metrics between the interleaving distance (which
is complete on finitely-presented modules) and the matching distance on fibered
bar codes. More precisely, we show that: for a finitely-presented
multiparameter module $M$ there is a neighbourhood of $M$, in the interleaving
distance $d_I$, for which the matching distance, $d_0$, satisfies the following
bi-Lipschitz inequalities $\frac{1}{34}d_I(M,N) \leq d_0(M,N) \leq d_I(M,N)$
for all $N$ in this neighbourhood about $M$. As a consequence no other module
in this neighbourhood has the same fibered bar code as $M$.
