Kadison proved that the diagonal entries of a projection must satisfy an integrality condition, but where that integer comes from remained largely a mystery. Ng, Zhang, and I proved that a necessary and sufficient condition for positive operators to be infinite sums of projections also involves an integrality condition. I will show how these two cases are related and how in both cases the integers involved are Fredholm indices of a certain natural frame transform. This leads also to an alternate approach to the sufficiency part in Kadison's theorem and to a necessary condition for the open problem of when an operator is the sum of finitely many projections.