In this paper, we present the analysis and results of a direct measurement of the cosmic-ray proton spectrum with the CALET instrument onboard the International Space Station, including the detailed assessment of systematic uncertainties. The observation period used in this analysis is from October 13, 2015 to August 31, 2018 (1054 days). We have achieved the very wide energy range necessary to carry out measurements of the spectrum from 50 GeV to 10 TeV covering, for the first time in space, with a single instrument the whole energy interval previously investigated in most cases in separate subranges by magnetic spectrometers (BESS-TeV, PAMELA, and AMS-02) and calorimetric instruments (ATIC, CREAM, and NUCLEON). The observed spectrum is consistent with AMS-02 but extends to nearly an order of magnitude higher energy, showing a very smooth transition of the power-law spectral index from -2.81 +- 0.03 (50--500 GeV) neglecting solar modulation effects (or -2.87 +- 0.06 including solar modulation effects in the lower energy region) to -2.56 +- 0.04 (1--10 TeV), thereby confirming the existence of spectral hardening and providing evidence of a deviation from a single power law by more than 3 sigma.
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