= Solution
\b[Correction of the printed limits:] ordinary positive <polymer contour length> requires integration from $R_{\rm in}$ to $R_{\rm out}$, because $R_{\rm out}>R_{\rm in}$ and the ring count is positive. For example, taking $n(R)=1$, $R_{\rm out}=2$ and $R_{\rm in}=1$ makes the printed outward-to-inward integral negative. Thus its sign cannot be retained as a physical length.
For hexagonally packed strands choose axial ring spacing $s$ and radial layer spacing $\sqrt3s/2$. With $n=h/s$ and radial layer density $2/(\sqrt3s)$,
$$
L=\frac{2}{\sqrt3s}\int_{R_{\rm in}}^{R_{\rm out}}2\pi R\frac hs\,dR
=\frac{2\pi h}{\sqrt3s^2}(R_{\rm out}^2-R_{\rm in}^2).
$$
The same continuum replacement in <3/b> gives
$$
\Delta G_{\rm elastic}
=\frac{2\pi l_pk_BT h}{\sqrt3s^2}\ln\frac{R_{\rm out}}{R_{\rm in}},\qquad
R_{\rm in}^2=R_{\rm out}^2-\frac{\sqrt3s^2L}{2\pi h}.
$$
Eliminating the inner radius yields
$$
\boxed{\Delta G_{\rm elastic}
=\frac{\pi l_pk_BT h}{\sqrt3s^2}
\ln\left(\frac{R_{\rm out}^2}{R_{\rm out}^2-\sqrt3s^2L/(2\pi h)}\right).}
$$
The resisting packaging <force> is the marginal <free energy> per added <polymer contour length>, at fixed $h$, $s$ and $R_{\rm out}$:
$$
\boxed{F_L=\left(\frac{\partial\Delta G_{\rm elastic}}{\partial L}\right)_{h,s,R_{\rm out}}
=\frac{l_pk_BT}{2R_{\rm in}^2}
=\frac{l_pk_BT}{2\left(R_{\rm out}^2-\sqrt3s^2L/(2\pi h)\right)}.}
$$
The <force> opposing insertion is positive; its contribution to outward ejection has the same magnitude. The domain is $0\leq L<2\pi hR_{\rm out}^2/(\sqrt3s^2)$. The divergence as the inner hole closes reflects the singular curvature of ideal rings and marks breakdown of the continuum model. If spacing also relaxes, the full <free energy> must include interstrand interactions before optimization; simply varying $s$ in this bending-only formula is insufficient.
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