  The ongoing experimental efforts to measure the hyperfine transition in
muonic hydrogen prompt an accurate evaluation of the proton-structure effects.
At the leading order in $\alpha$, which is $O(\alpha^5)$ in the hyperfine
splitting (hfs), these effects are usually evaluated in a data-driven fashion,
using the empirical information on the proton electromagnetic form factors and
spin structure functions. Here we perform a first calculation based on the
baryon chiral perturbation theory (B$\chi$PT). At leading orders it provides a
prediction for the proton polarizability effects in hydrogen (H) and muonic
hydrogen ($\mu$H). We find large cancellations among the various contributions
leading to, within the uncertainties, a zero polarizability effect at leading
order in the B$\chi$PT expansion. This result is in significant disagreement
with the current data-driven evaluations. The small polarizability effect
implies a smaller Zemach radius $R_\mathrm{Z}$, if one uses the well-known
experimental $1S$ hfs in H or the $2S$ hfs in $\mu$H. We, respectively, obtain
$R_\mathrm{Z}(\mathrm{H}) = 1.010(9)$ fm, $R_\mathrm{Z}(\mu\mathrm{H}) =
1.040(33)$ fm. The total proton-structure effect to the hfs at $O(\alpha^5)$ is
then consistent with previous evaluations; the discrepancy in the
polarizability is compensated by the smaller Zemach radius. Our recommended
value for the $1S$ hfs in $\mu\text{H}$ is $182.640(18)\,\mathrm{meV}.$
