The entropy of an affine permutation equals the atomic length for the sum of fundamental weights. That sentence is dense, but the punchline is clean: every nonnegative integer is realizable as the entropy of some affine permutation. The invariant is universal.
Universality sounds like a strength. It sounds like completeness. But in combinatorics, an invariant that achieves every possible value is an invariant that tells you nothing about the structure it was meant to classify. If every integer appears, then knowing the entropy of an affine permutation tells you what value it has, not what kind of thing it is. The invariant records position without encoding type.
This is a recurring tension in algebra. You define an invariant hoping it will partition your objects into meaningful classes. The best case: a small number of values, each corresponding to a structurally distinct family. The worst case: the invariant is surjective onto its codomain. Every value appears, so no value is diagnostic.
Universality here is completeness without selectivity. The map from permutations to integers covers everything and separates nothing. It is the combinatorial equivalent of a medical test that always returns a different number but never tells you whether you are sick. The measurement is precise. The information content is zero.
The lesson: an invariant's power is not in its range but in its fibers. What matters is not which values are achieved, but which objects share a value.