  In this paper, we introduce the notion of augmentation for polytopes and use
it to show the error in two presumptions that have been key in arriving at
over-reaching/over-scoped claims of "impossibility" in recent extended
formulations (EF) developments. One of these presumptions is that: "If
Polytopes P and Q are described in the spaces of variables x and y
respectively, and there exists a linear map x=Ay between the feasible sets of P
and Q, then Q is an EF of P". The other is: "(An augmentation of Polytope A
projects to Polytope B) ==> (The external descriptions of A and B are
related)". We provide counter-examples to these presumptions, and show that in
general: (1) If polytopes can always be arbitrarily augmented for the purpose
of establishing EF relations, then the notion of EF becomes
degenerate/meaningless in some cases, and that: (2) The statement: "(Polytope B
is the projection of an augmentation of Polytope A) ==> (Polytope B is the
projection of Polytope A)" is not true in general (although, as we show, the
converse statement, "(B is the projection of A) ==> (B is the projection of
every augmentation of A)", is true in general). We illustrate some of the ideas
using the minimum spanning tree problem, as well as the "lower bounds"
developments in Fiorini et al. (2011; 2012), in particular.
