Centaurus A (Cen A) is the nearest radio galaxy discovered as a very-high-energy (VHE; 100 GeV-100 TeV) $gamma$-ray source by the High Energy Stereoscopic System (H.E.S.S.). It is a faint VHE $gamma$-ray emitter, though its VHE flux exceeds both the extrapolation from early Fermi-LAT observations as well as expectations from a (misaligned) single-zone synchrotron-self Compton (SSC) description. The latter satisfactorily reproduces the emission from Cen A at lower energies up to a few GeV. New observations with H.E.S.S., comparable in exposure time to those previously reported, were performed and eight years of Fermi-LAT data were accumulated to clarify the spectral characteristics of the $gamma$-ray emission from the core of Cen A. The results allow us for the first time to achieve the goal of constructing a representative, contemporaneous $gamma$-ray core spectrum of Cen A over almost five orders of magnitude in energy. Advanced analysis methods, including the template fitting method, allow detection in the VHE range of the core with a statistical significance of 12$sigma$ on the basis of 213 hours of total exposure time. The spectrum in the energy range of 250 GeV-6 TeV is compatible with a power-law function with a photon index $Gamma=2.52pm0.13_{mathrm{stat}}pm0.20_{mathrm{sys}}$. An updated Fermi-LAT analysis provides evidence for spectral hardening by $DeltaGammasimeq0.4pm0.1$ at $gamma$-ray energies above $2.8^{+1.0}_{-0.6}$ GeV at a level of $4.0sigma$. The fact that the spectrum hardens at GeV energies and extends into the VHE regime disfavour a single-zone SSC interpretation for the overall spectral energy distribution (SED) of the core and is suggestive of a new $gamma$-ray emitting component connecting the high-energy emission above the break energy to the one observed at VHE energies.
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