  In this paper we introduce a collection of powerful statistical comparison
results. We first present the results that we obtained while developing a
general comparison concept. After that we introduce a separate lifting
procedure that is a comparison concept on its own. We then show how in certain
scenarios the lifting procedure basically represents a substantial upgrade over
the general strategy. We complement the introduced results with a fairly large
collection of numerical experiments that are in an overwhelming agreement with
what the theory predicts. We also show how many well known comparison results
(e.g. Slepian's max and Gordon's minmax principle) can be obtained as special
cases. Moreover, it turns out that the minmax principle can be viewed as a
single max principle as well. The range of applications is enormous. It starts
with revisiting many of the results we created in recent years in various
mathematical fields and recognizing that they are fully self-contained as their
starting blocks are specialized variants of the concepts introduced here.
Further upgrades relate to core comparison extensions on the one side and more
practically oriented modifications on the other. Those that we deem the most
important we discuss in several separate companion papers to ensure preserving
the introductory elegance and simplicity of what is presented here.
