A cautionary note on operator bases in the exponential Stueckelberg parametrisation: Non-Noetherian completions and algorithmic hazards

In the standard formulation of chiral Lagrangians and Stueckelberg extensions, the use of the unitary field U = e i π / f together with explicit power counting in 1/ f guarantees a finite operator basis at any fixed mass dimension. This work does not challenge that well-established physical fact. Instead, we investigate a subtle algebraic phenomenon that arises when one adopts the logarithmic (exponential) parametrisation π (with f absorbed) in automated operator enumeration algorithms. We show that the expansion of the Maurer–Cartan form Ω = U − 1 d U in powers of π generates an infinite tower of local densities that remain independent in the integration-by-parts (IBP) quotient when treated as elements of a filtered completion of the local jet algebra. This non-Noetherian behaviour signals a potential failure mode for polynomial Hilbert-series methods if the allowed function class is not explicitly declared. Our main contribution is a systematic diagnostic tool — a separating-functional lemma and an explicit SU(2) witness — that determines when such infinite towers survive IBP reduction. This diagnostic serves as a practical warning for developers of automated EFT codes: the finiteness of an operator basis depends on the choice of field parametrisation and the function class fed into the algorithm.