  The now famous Isothermal Theorem was introduced in a Nature Communications
article in 2005 by Nowak and Liebermann and has since been the founding
document of the rich field of evolutionary graph theory. Unfortunately, the
Isothermal Theorem has never been proven completely. The main argument, that
the projection of the graph dynamics is a Markovian Birth-and-Death- Process,
is not applicable in general and leaves the question open for the case, when
the projection is not Markov. To complete the proof, we first generalise the
model by Liebermann and Nowak, by introducing a non-uniform selection policy to
select an individual for procreation. Then, we use a martingale argument to
prove that selecting with a specific selection policy related to the weight
matrix of the underlying population graph still gives rise to Moran fixation
probability, even outside of the framework of the original Isothermal Theorem.
Our proof includes and completes the proof of the Isothermal Theorem. We follow
up with a small numerical study that shows that the set of spatial Moran
Processes with Moran fixation probability is even richer than previously
understood. The initial condition and the selection policy play an important
role in this.
