Halliday & Matthiessen (2014: 537):
There is one other type of projection, where the projected clause is not being projected by a verbal or mental process with Sayer or Senser, or by a verbal or mental process noun in a metaphorical nominal group, but comes, as it were, ready packaged in projected form. We refer to this type as a fact. For example:
The irony is even further compounded by the fact [[that while every people must have those [[[who say, || ‘Here I stand,’]]] the fact is [[[Okonkwo loses a child to the forces of Christianity, || and Ezeulu loses his community to Christianity]]] ]].
The fact [[that fourteen of the original eighteen Julian Ashton nudes still decorate the Marble Bar’s walls]], perchance contributed to the bar winning Australian Playboy’s survey for Best Bar in Australia in 1986.
The fact [[that Lear never even alluded to that at the end]] is a sign [[that he didn’t learn very much through the course of the play]].
He’s trapped by the fact [[that the river flows south]].
Here a ‘fact’ noun serves as the Head/Thing of a nominal group with a projected clause as Postmodifier/Qualifier. Consider That Caesar was dead was obvious to all. Here that Caesar was dead is certainly a projection; but there is no process of saying or thinking which projects it. Its status is simply that of a fact; and it can indeed function as Qualifier to the noun fact, e.g. the fact that Caesar was dead was obvious to all.