  While decomposition of one-parameter persistence modules behaves nicely, as
demonstrated by the algebraic stability theorem, decomposition of
multiparameter modules is well known to be unstable in a certain precise sense.
Until now, it has not been clear that there is any way to get around this and
build a meaningful stability theory for multiparameter module decomposition, as
naive attempts to do this tend to fail. We introduce tools and definitions, in
particular $\epsilon$-refinements and the $\epsilon$-erosion neighborhood of a
module, to make sense of the question of how to build such a theory. Having
built this foundation, we define the $\epsilon$-pruning of a module, which is a
new invariant acting like a ``refined barcode'' that shows great promise to
extract features from a module by approximately decomposing it.
  The main theorem of the paper can be interpreted as a generalization of the
algebraic stability theorem to multiparameter modules up to a factor of $2r$,
where $r$ is the maximal pointwise dimension of one of the modules. Another
interpretation is that the pruning of a module $M$ detects all summands
appearing in modules in a neighborhood of $M$. Furthermore, we show that the
factor $2r$ appearing in the theorem is close to optimal. We also define a
distance that we conjecture to be a stable bottleneck-type distance, which has
long been a missing piece in multipersistence theory. Finally, we discuss
whether the main theorem can be improved for modules that decompose into
pointwise low-dimensional summands, and pose a conjecture phrased purely in
terms of basic linear algebra and graph theory that seems to capture the
difficulty of doing this. We also show that this conjecture is relevant for
other areas of multipersistence, like the complexity of approximating the
interleaving distance, and recent applications of homological algebra to
multipersistence.
