  This paper provides theorems aimed at shedding light on issues in the
foundations of quantum mechanics. These theorems can be used to propose new
interpretations to the theory, or to better understand, evaluate and improve
current interpretations. Some of these applications include: (1) A proof of the
existence of pilot-wave theories that are fully equivalent to standard quantum
mechanics in a path-wise sense. This equivalence is stronger than what is
entailed from the more traditional requirements of equivariance, or good mixing
properties, and is necessary to assure proper correlations across time and
proper records of the past. (2) A proposal for a minimalistic ontology for
non-collapse quantum mechanics, in which Born's rule provides the proper
predictions.(3) The observation of a close relationship between Born's rule and
a version of the superposition principle.
