  The lift of K-theoretic D-brane charge to M-theory was recently hypothesized
to land in Cohomotopy cohomology theory. To further check this "Hypothesis H",
here we explicitly compute the constraints on fractional D-brane charges at
ADE-orientifold singularities imposed by the existence of lifts from
equivariant K-theory to equivariant Cohomotopy theory, through Boardman's
comparison homomorphism. We check the relevant cases and find that this
condition singles out precisely those fractional D-brane charges which do not
take irrational values, in any twisted sector. Given that the possibility of
irrational D-brane charge has been perceived as a paradox in string theory, we
conclude that Hypothesis H serves to resolve this paradox.
  Concretely, we first explain that the Boardman homomorphism, in the present
case, is the map from the Burnside ring to the representation ring of the
singularity group given by forming virtual permutation representations. Then we
describe an explicit algorithm that computes the image of this comparison map
for any finite group. We run this algorithm for binary Platonic groups, hence
for finite subgroups of SU(2); and we find explicitly that for the three
exceptional subgroups and for the first few cyclic and binary dihedral
subgroups the comparison morphism surjects precisely onto the sub-lattice of
the real representation ring spanned by the non-irrational characters.
