  In this study, we introduce a transformative, automated framework for
classifying basis invariants in generic field theories. Utilising a novel
ring-diagram methodology accompanied by the well-known Cayley-Hamilton theorem,
our approach uniquely enables the identification of basic invariants and their
CP-property characterisation. Critically, our framework also unveils previously
concealed attributes of established techniques reliant on the
Hilbert-Poincar\'e series and its associated Plethystic logarithm. This
paradigm shift has broad implications for the deeper understanding and more
accurate classification of CP invariants in generic field theories.
