  We find and analyze the exact time-dependent potential energy surface driving
the proton motion for a model of cavity-induced suppression of proton-coupled
electron-transfer. We show how, in contrast to the polaritonic surfaces, its
features directly correlate to the proton dynamics and discuss
cavity-modifications of its structure responsible for the suppression. The
results highlight the interplay between non-adiabatic effects from coupling to
photons and coupling to electrons, and suggest caution is needed when applying
traditional dynamics methods based on polaritonic surfaces.
