Origin of the sum rule change when composite excitons are bosonized

We here present a direct way to show why a factor 1/2 between the inverse lifetime and the sum of scattering rates has to be missed when excitons are replaced by elementary bosons. This striking factor, which physically comes from the exciton composite nature, can be mathematically assigned to the closure relation for exciton states which exists in spite of the overcompleteness of the set they form, and which differs from the one for elementary bosons. This should challenge the large amount of physicists who still trust bosonization.