Although this section is limited to one dimension only, the framework is general: it outlines in principle a systematic procedure for obtaining both the dimensionality and explicit realizations of an operator basis for an EFT.

Sections \[sec:intuitive\_calc\] and \[sec:su2\] explore the possibility of summing the series and obtaining an all-orders result for the EFT considered here. Section \[sec:intuitive\_calc\] relies heavily on the formalism developed above, and we show that while the summed series is easily obtainable when accounting for only IBP or EOM, taken together the problem becomes considerably complicated. We present a sum formula for the Hilbert series for general $N$, for which we are able to perform the summation for relatively low values of $N$. Section \[sec:su2\] exploits a connection with the representation theory of $SL(2,{\mathbb C})$ that is present in this theory. Using this formalism, we are able to obtain the closed form of the Hilbert series for general $N$.

We go on to explore the analytic structure of the Hilbert series in section \[sec:poles\]. This reveals interesting consistency conditions and recursion relations connecting different Hilbert series. We conclude in section \[sec:summary\] with a discussion aimed at highlighting the physical aspects of this work and the nature 