Mathematical proof concerning the additivity problem of nonlinear normalized citation counts

Abstract: The issue of whether nonlinear normalized citation counts can be added is critically important in scientometrics because it touches upon the theoretical foundation of underlying computation in the field. In this paper, we provide rigorous mathematical proofs for the key theorems underlying this fundamental issue. Based on these proofs, we ultimately arrive at the following conclusion: a nonlinear normalization method for citation counts must be a non-equidistant transformation; consequently, the resulting nonlinear normalized citation counts are no longer equidistant and therefore cannot be added. Furthermore, because our mathematical proofs are established over the real number domain, we also derive a more general conclusion that is applicable to data transformations over the real number domain across various scientific fields: a nonlinear transformation becomes a non-equidistant transformation only if it satisfies a certain regularity condition, for example, if this nonlinear transformation is a continuous function, monotonic over a certain interval, or its domain is restricted to rational numbers. In such cases, the resulting nonlinear data are no longer equidistant and therefore cannot be added. This general conclusion can be broadly applied to various linear and nonlinear transformation problems, which offers significant insights for addressing the misuse of nonlinear data.