= Solution
Use the elastic-rod bending functional for an abstract inextensible <worm-like chain>,
$$
G_b=\frac{\kappa}{2}\int C(s')^2\,ds',\qquad\kappa=l_pk_BT.
$$
A circular ring of radius $R$ has <curvature> $1/R$ and length $2\pi R$, hence <energy> $\pi\kappa/R$. With $n(R_i)$ rings of each radius, add their energies:
$$
\boxed{G_b=\pi l_pk_BT\sum_i\frac{n(R_i)}{R_i}.}
$$
For a cylindrical annulus of height $h$, choose neighboring rings separated vertically by $s$. Then $n=h/s$, while adjacent radial layers are separated by $\sqrt3s/2$. Replacing the radial sum by its continuum integral gives
$$
L=\frac{2}{\sqrt3s}\int_{R_{\rm in}}^{R_{\rm out}}2\pi R\frac hs\,dR
=\frac{2\pi h}{\sqrt3s^2}(R_{\rm out}^2-R_{\rm in}^2),
$$
and
$$
G_b=\frac{2\pi l_pk_BTh}{\sqrt3s^2}\int_{R_{\rm in}}^{R_{\rm out}}\frac{dR}{R}
=\frac{2\pi l_pk_BTh}{\sqrt3s^2}\ln\frac{R_{\rm out}}{R_{\rm in}}.
$$
Eliminate the inner radius to obtain
$$
\boxed{G_b=\frac{\pi l_pk_BTh}{\sqrt3s^2}
\ln\!\left[\frac{R_{\rm out}^2}{R_{\rm out}^2-\sqrt3Ls^2/(2\pi h)}\right].}
$$
The dimensions are <energy>, and the expression is positive for $0<L<2\pi hR_{\rm out}^2/(\sqrt3s^2)$. It tends to zero with $L$ and diverges as the inner hole closes in this continuum model. The divergence signals the growing cost of forcing an inextensible chain to zero radius; it is not a physical prediction at arbitrarily small molecular scales. Interstrand, hydration and configurational contributions have been excluded, and disconnected ideal rings represent local <curvature> without modeling the connections between them.
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