The MINERvA experiment observes an excess of events containing electromagnetic showers relative to the expectation from Monte Carlo simulations in neutral-current neutrino interactions with mean beam energy of 4.5 GeV on a hydrocarbon target. The excess is characterized and found to be consistent with neutral-current neutral pion production with a broad energy distribution peaking at 7 GeV and a total cross section of 0.26 +- 0.02 (stat) +- 0.08 (sys) x 10^{-39} cm^{2}. The angular distribution, electromagnetic shower energy, and spatial distribution of the energy depositions of the excess are consistent with expectations from neutrino neutral-current diffractive neutral pion production from hydrogen in the hydrocarbon target. These data comprise the first direct experimental observation and constraint for a reaction that poses an important background process in neutrino oscillation experiments searching for muon neutrino to electron neutrino oscillations.
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