  The polarized cross section ratio $\sigma_{LT'}/\sigma_{0}$ from hard
exclusive $\pi^{-} \Delta^{++}$ electroproduction off an unpolarized hydrogen
target has been extracted based on beam-spin asymmetry measurements using a
10.2 GeV / 10.6 GeV incident electron beam and the CLAS12 spectrometer at
Jefferson Lab. The study, which provides the first observation of this channel
in the deep-inelastic regime, focuses on very forward-pion kinematics in the
valence regime, and photon virtualities ranging from 1.5 GeV$^{2}$ up to 7
GeV$^{2}$. The reaction provides a novel access to the $d$-quark content of the
nucleon and to $p \rightarrow \Delta^{++}$ transition generalized parton
distributions. A comparison to existing results for hard exclusive $\pi^{+} n$
and $\pi^{0} p$ electroproduction is provided, which shows a clear impact of
the excitation mechanism, encoded in transition generalized parton
distributions, on the asymmetry.
