Chi^2 distribution clarifications

[DWesl]

The paper referencing this code mentions that
X² = s_prior^T B^-1 s_prior + z^T R^-1 z
is distributed as a χ² variable on ν = N_observations + N_fluxes. This is true.

The paper then goes on to generate an ensemble of pseudo-observations, calculate X²/ν for each member, and states that the expected value of this latter quantity is one. This is also true.

The difficulty I am having is that I can decompose X² into two random variables:
X_flux² = s_prior^T B^-1 s_prior
and
X_obs² = z^T R^-1 z,
which are distributed as χ² variables on N_fluxes and N_observations, respectively. The ensemble in the paper generates multiple realizations of the second variable, but I only see one realization of the first.
E(X² / ν) is still one;
E(E(X² | X_flux²)) is also one;
E(X² | X_flux²) is a function of X_flux² and is what I think the paper has.

Is this an accurate description of what is going on? Is this a known and intended behavior of this package? It is entirely possible I just missed this section of the paper while reading. If that is th case, a pointer to the section of the paper where this is described would be welcome.