This paper reports measurements of final-state proton multiplicity, muon and proton kinematics, and their correlations in charged-current pionless neutrino interactions, measured by the T2K ND280 near detector in its plastic scintillator (C$_8$H$_8$) target. The data were taken between years 2010 and 2013, corresponding to approximately 6$times10^{20}$ protons on target. Thanks to their exploration of the proton kinematics and of kinematic imbalances between the proton and muon kinematics, the results offer a novel probe of the nuclear-medium effects most pertinent to the (sub-)GeV neutrino-nucleus interactions that are used in accelerator-based long-baseline neutrino oscillation measurements. These results are compared to many neutrino-nucleus interaction models which all fail to describe at least part of the observed phase space. In case of events without a proton above a detection threshold in the final state, a fully consistent implementation of the local Fermi gas model with multinucleon interactions gives the best description of the data. In the case of at least one proton in the final state the spectral function model agrees well with the data, most notably when measuring the kinematic imbalance between the muon and the proton in the plane transverse to the incoming neutrino. A clear indication of existence of multinucleon interactions is observed. The effect of final-state interactions is also discussed.
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