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Using data taken at 23 center-of-mass energies between 4.0 and 4.6 GeV with the BESIII detector at the BEPCII collider and with a total integrated luminosity of approximately $15$ fb$^{-1}$, the process $e^+e^-\to 2(p \bar{p})$ is studied for the first time. The Born cross sections for $e^+e^-\to 2(p \bar{p})$ are …