We have measured new observables based on the final state kinematic imbalances in the mesonless production of $ u_mu+Arightarrowmu^-+p+X$ in the $text{MINER} utext{A}$ tracker. Components of the muon-proton momentum imbalances parallel ($delta p_mathrm{Ty}$) and perpendicular($delta p_mathrm{Tx}$) to the momentum transfer in the transverse plane are found to be sensitive to the nuclear effects such as Fermi motion, binding energy and non-QE contributions. The QE peak location in $delta p_mathrm{Ty}$ is particularly sensitive to the binding energy. Differential cross sections are compared to predictions from different neutrino interaction models. The Fermi gas models presented in this study cannot simultaneously describe features such as QE peak location, width and the non-QE events contributing to the signal process. Correcting the GENIEs binding energy implementation according to theory causes better agreement with data. Hints of proton left-right asymmetry are observed in $delta p_mathrm{Tx}$. Better modeling of the binding energy can reduce bias in neutrino energy reconstruction and these observables can be applied in current and future experiments to better constrain nuclear effects.
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