  We calculate the inelastic $2\to3$ one-loop amplitude for the scattering of
two point-like, spinless objects with generic masses involving the additional
emission of a single graviton. We focus on the near-forward, or classical,
limit. Our results include the leading and subleading orders in the soft-region
expansion, which captures all non-analytic contributions in the transferred
momentum and in the graviton's frequency. This allows us to check the first
constraint arising from the inelastic exponentiation put forward in Refs.
2107.12891, 2112.07556, 2210.12118 and to calculate the $2\to3$ one-loop matrix
element of the $N$-operator, linked to the $S$-matrix by $S = e^{iN}$, showing
that it is real, classical and free of infrared divergences. We discuss how our
results feature in the calculation of the $\mathcal O(G^3)$ corrections to the
asymptotic waveform.
