Identifying the countries
An important realisation, along with recognising maps, is that
not all of them are full countries (some of them are just parts of countries), and some of them are rotated or reflected to be not in their usual orientation.
Exhibit 1 - two parts
1a is CANADA (western part).
1b is BRAZIL (northern part).
Exhibit 2 - four parts
2a is CUBA (western part).
2b is TURKEY (northern part) with ANKARA in red.
2c is PAKISTAN (southern part).
2d is BRAZIL (southern part, flipped top to bottom).
Exhibit 3 - one part
3a is AUSTRALIA (eastern part, flipped right to left) with CANBERRA in red.
Exhibit 4 - twelve parts
4a is MALI (western part).
4b is GREECE (northern part).
4c is GREECE (southern part).
4d is PORTUGAL (southern part, flipped top to bottom) with LISBON in red.
4e is ITALY (western part, flipped right to left) with MILAN in red.
4f is ALGERIA (northern part, flipped top to bottom) with ALGIERS in red.
4g is FRANCE (northern part).
4h is NIGERIA (western part) with LAGOS in red.
4i is HONDURAS (southern part).
4j is HONDURAS (northern part).
4k is ICELAND (northern part).
4l is AUSTRALIA (southern part, flipped top to bottom) with PERTH in red.
(thanks to @Stiv for some help here!)
The OP says that "Exhibits 1, 2, and 3 demonstrate equivalency", so maybe we're meant to get:
CAN from Canada, BRA from Brazil;
C from CUBA, A from Ankara, N from Pakistan, BRA from Brazil (again);
CANBERRA from Canberra.
Then the meaning of Exhibit 4 might be along the lines of:
MA from MALI.
G or GR or GRE from GREECE.
E or CE or ECE from GREECE.
N or ON or BON from LISBON.
M or MI or MIL from MILAN.
A or AL or ALG from ALGIERS.
F from FRANCE.
LAG from LAGOS.
S from HONDURAS.
H from HONDURAS.
I from ICELAND.
P from PERTH.
It looks like we're getting
something FLAGSHIP, but I can't figure out the early parts. Maybe the red
+ next to Milan and Lagos in the maps indicates that we should take more than half of those words, so it's MIL and LAG. From the early parts it looks something like MAGHREB but not quite.
I suppose the answer might be
ALGIERS, if that city can be seen as the flagship of the Maghreb. But probably not.
Some older guesses
Maybe each country is represented by a code, such as its two-letter internet domain or its telephone prefix. Then
taking just half of a country could mean taking just one letter or number from such a code. But that wouldn't explain the significance of flipping some of the half-countries,
so it's probably something else. Maybe codes aren't involved at all and there is some pattern in the given countries which points towards whatever the final one is to be identified. Another option could be
The interesting thing here is that
CANBRA is almost CANBERRA,
but it might be a coincidence as I can't make much sense of the rest.
The red dots also need to be taken into account:
some but not all of them are capital cities, some of them have a red + or - next to the dot but some don't.
The cities featured are
Ankara; Canberra; Lisbon, Milan, Algiers, Lagos. I couldn't spot any pattern in these yet.