  Based upon the resistor analogy and using the ideal loop gas
approximation(ILGA) it is shown that only pending loops reduce the modulus of
an otherwise perfect network made of monodisperse strands and junctions of
identical functionality. Thus, the cycle rank of the network with pending
structures removed (cyclic and branched) is sufficient to characterize modulus,
if the resistor analogy can be employed. It is further shown that it is
impossible to incorporate finite cycles into a polymer network such that
individual network strands are at equilibrium conformations while maintaining
simultaneously a force balance at the junctions. Therefore, the resistor
analogy provides only an approximation for the phantom modulus of networks
containing finite loops. Improved approaches to phantom modulus can be
constructed from considering a force balance at the junctions, which requires
knowledge of the distribution of cross-link fluctuations in imperfect networks.
Assuming loops with equilibrium conformations and a force balance at all loop
junctions, a lower bound estimate for the phantom modulus,
$G_{\text{ph}}\approx\left(\xi-c_{\text{f}}L_{1}\right)kT/V$ is obtained within
the ILGA for end-linked model networks and in the limit of $L_{1}\ll\xi$. Here,
$L_{1}$ is the number of primary ("pending") loops, $\xi$ the cycle rank of the
network, $k$ the Boltzmann constant, $V$ the volume of the sample, and $T$ the
absolute temperature. $c_{\text{f}}$ is a functionality dependent coefficient
that is $\approx2.56$ for junction functionality $f=3$ and $\approx3.06$ for
$f=4$, while it converges quickly towards $\approx4.2$ in the limit of large
$f$. Further corrections to phantom modulus beyond finite loops are addressed
briefly.
