A Discrete Operator for the Formalization of Valence States in Closed-Shell Configurations

The valences of representative elements are usually presented as empirical rules associated with the periodic table and qualitative electronic trends. Although these valences are routinely used in chemical formulation, teaching, and heuristic reasoning, they are rarely expressed through an explicit mathematical structure. In this work, a simple discrete operator is proposed that allows the generation of valence values corresponding to closed-shell configurations (paired electrons) of main-group elements from the number of valence electrons. The operator reproduces the known patterns of groups 1, 2, and 13–17 and provides a compact formalization of the concept of combining capacity in classical chemistry. The aim of the model is not to replace quantum theories of bonding, but to offer a heuristic and algorithmic tool useful for didactic and computational applications.