Lift of fractional D-brane charge to equivariant Cohomotopy theory

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