Why is $d^2 U=0$ for a thermal reservoir? In section 6.1 of Herbert B. Callen's Thermodynamics and an Introduction to Thermostatistics, the author sets out to derive extremum principles for the Legendre transformed representations (i.e. thermodynamic potentials), analogous to the entropy maximum and energy minimum principles from earlier in the book. His argument starts as follows:

For definiteness consider a composite system in contact with a thermal reservoir. Suppose further that some internal constraint has been removed. We seek the mathematical condition that will permit us to predict the equilibrium state. For this purpose we first review the solution of the problem by the energy minimum principle.
In the equilibrium state the total energy of the composite system-plus-reservoir is minimum:
$$d(U + U^r) = 0\tag{6.1}$$
and
$$d^2(U + U^r) = d^2 U > 0\tag{6.2}$$
subject to the isentropic condition
$$d(S + S^r) = 0.\tag{6.3}$$
The quantity $d^2U^r$ has been put equal to zero in equation 6.2 because $d^2U^r$ is a sum of products of the form
$$\frac{\partial^2U^r}{\partial X^r_j \partial X^r_k} dX^r_j dX^r_k$$
which vanish for a reservoir (the coefficient varying as the reciprocal of the mole number of the reservoir).

I don't understand this last part. Why is $d^2U^r = 0$? What is the author referring to as "the coefficient", why does it vary as the reciprocal of the mole number, and why does that imply that the second order differentials vanish?
I have not been able to find an explanation from what I've read in the previous chapters. It seems uncharacteristic for Callen to throw out a statement like this without proper motivation. He normally carefully derives everything from first postulates, which is a big reason why I've enjoyed the book so far. But maybe I'm just missing something obvious.
 A: A thermal reservoir's internal energy $U^r$ depends on only one parameter, namely its  temperature, i.e., $U^r=f(T^r)$ and is independent of all the other variables the system to which it is attached may have; furthermore every process in it is reversible $dU^r = T^rdS^r$.
A: I think I have managed to resolve this in a relatively satisfying way with inspiration from a question linked in a comment by nosuchthingasmagic. It uses only definitions encountered earlier in the book, but it does not seem to be the argument that the author had in mind. I would therefore still appreciate if someone could enlighten me as to what Callen meant in the quoted statement in my original question. Anyway, here goes.
On page 106 Callen defines a thermal reservoir:

A thermal reservoir is defined as a reversible heat source that is so large that any heat transfer of interest does not alter the temperature of the thermal reservoir.

This tells us, in particular, that
$$\frac{dT}{\delta Q} = 0$$
for the thermal reservoir.
The definition also mentions that it is a "reversible heat source", which is defined on page 104:

Reversible heat sources are defined as systems enclosed by rigid impermeable walls and characterized by relaxation times sufficiently short that all processes of interest within them are essentially quasi-static.

Hence $dV = dN = 0$ and $\delta Q = T dS$ holds. The second differential of the internal energy of the thermal reservoir is therefore simply
$$d^2 U = \left(\frac{\partial^2 U}{\partial S^2}\right)_{V,N} (dS)^2 = \left(\frac{\partial T}{\partial S}\right)_{V,N} (dS)^2 = T \left(\frac{dT}{\delta Q}\right)_{V,N} (dS)^2 = 0.$$
