When a complementarity in the neutrino and the quark mixing meets a parameter symmetry and its implications to the unitarity

We present a complementarity that addresses relationships among the parameters in the neutrino and the quark mixing matrix, use it to estimate the size of the uncertainty among the elements in the matrix and address its implications to the unitarity of the quark mixing matrix and Wolfenstein parameterization and the tension in the first row. First, we describe how a complementarity with a phase being introduced as an extra parameter can be held in the nine independent schemes of parameterizing the matrix introducing a discrete parameter symmetry within a certain size of uncertainty and how it can be related to a combination of sine functions. With that, for the first time, we describe a method that we can use to constrain the size of the uncertainty associated with the parameters, not the central values, complementing that among the diagonal elements in the neutrino mixing matrix. Then we do the same for the quark sector and discuss its implication in the relation to the size of the uncertainty among the elements. Seeing that our estimation is larger than that was reported by running the global fit in the quark sector, our result could be an indication that we may need to be cautious when addressing the tension in the first row of the matrix in the quark sector and when running global fit to constrain the size of the uncertainty, where Wolfenstein parameterization, one that is not unitarity guaranteed, is used, as opposed to the combination of the three rotational matrix. Given that the size of the uncertainty for the individual diagonal element in the second and the third row, our result also could be an indication that we may need to wait until the size of uncertainty for the second and the third row goes down further before addressing the tension. It could be an opening of considering the possibility of a mixing between the neutrino and the quark sector too.