  We derive two new sum rules for the unpolarized doubly virtual Compton
scattering process on a nucleon, which establish novel low-$Q^2$ relations
involving the nucleon's generalized polarizabilities and moments of the
nucleon's unpolarized structure functions $F_1(x,Q^2)$ and $F_2(x,Q^2)$. These
relations facilitate the determination of some structure constants which can
only be accessed in off-forward doubly virtual Compton scattering, not
experimentally accessible at present. We perform an empirical determination for
the proton and compare our results with a next-to-leading-order chiral
perturbation theory prediction. We also show how these relations may be useful
for a model-independent determination of the low-$Q^2$ subtraction function in
the Compton amplitude, which enters the two-photon-exchange contribution to the
Lamb shift of (muonic) hydrogen. An explicit calculation of the
$\Delta(1232)$-resonance contribution to the muonic-hydrogen $2P-2S$ Lamb shift
yields $-1 \pm 1$ $\mathsf\mu$eV, confirming the previously conjectured
smallness of this effect.
