  Kadison's Pythagorean theorem (2002) provides a characterization of the
diagonals of projections with a subtle integrality condition. Arveson (2007),
Kaftal, Ng, Zhang (2009), and Argerami (2015) all provide different proofs of
that integrality condition. In this paper we interpret the integrality
condition in terms of the essential codimension of a pair of projections
introduced by Brown, Douglas and Fillmore (1973), or, equivalently of the index
of a Fredholm pair of projections introduced by Avron, Seiler, and Simon
(1994). The same techniques explain the integer occurring in the
characterization of diagonals of selfadjoint operators with finite spectrum by
Bownik and Jasper (2015).
