Let X be a Cantor space and let Q⊂Mfc(X) be a compact Choquet simplex of atomless full-support probability measures. We introduce the associated Affine Evaluation Map (AEM), which assigns to each clopen set A⊂X
Nearby in the stack
the affine function
A(μ)=μ(A),μ∈Q,
and study the geometric subset condition obtained by comparing these evaluation functions pointwise on
Q
. For a good geometric AEM we construct the ordered group
GQ=C(X,Z)/NQ,NQ={f∈C(X,Z):∫fdμ=0for every μ∈Q},
and show that it is a simple dimension group whose normalized state space is canonically
Q
. We prove that its order interval
[0,u]
is precisely the clopen scale and that
JQ=NQ,
where
JQ
is generated by the elementary relations
1A−1B
with
A=B
. We also show that the full stabilizer
HQ
has invariant-measure simplex exactly
Q
. Using the clopen-scale property and the Herman--Putnam--Skau realization theorem, we obtain a dimension-group proof that every good geometric AEM is realized by a minimal Cantor homeomorphism
T
with
Q=MT(X)
. For a Cantor minimal system we identify
GQ
with the classical dimension group modulo infinitesimals. We further distinguish goodness, fairness, ergodicity, and minimality of the measure stabilizer by explicit examples. Finally, we apply the AEM framework to minimal Cantor actions of countable amenable groups. If
Q=MG(X)
, then
Q
canonically defines a proper geometric AEM, and we clarify which parts of the preceding theory depend only on
Q
and which are specifically
Z
-dynamical. In particular,
Q
is good if and only if there exists a minimal homeomorphism