is useful to think of $H_{\br}(R_{\br}^{G_{\br}}/I_{\br},t)$ heuristically having the form $$H_{\br}(R_{\br}^{G_{\br}}/I_{\br},t) = \sum_{k=0}^{\infty}t^k \sum_{k_1+\dots+k_N = k}\big\{\text{conditions}\big\},
\label{eqn:heuristic_H_r}$$ where {conditions} abstractly denotes the conditions for counting independent degree $k$ polynomials in the module. Such conditions are encoded in the specific form of the ideal; for example, the consequence $k_i\le r_i$ noted above is reflected by the equations $\big(P_1^{(i)}\big)^{r_i+1}$ in the ideal of eq. . Third, because we sum over $\br$ to get the full Hilbert series, $H_N = \sum_{\br} \bu^{\br}H_{\br}$, and because, in general, $H_{\br}$ is a piecewise function of $\br$, it may prove most prudent to leave $H_{\br}$ as a sum formula of the form in eq. . We will encounter such a situation when faced with the general formula for $H_N$, eq. , the derivation of which we now turn to.

Computing the Hilbert series $\hnui$ {#sec:intuitive_calc}
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The aim of this section is to obtain a formula for the generalized Hilbert series, eq. , for the one-dimensional EFT of $N$ real scalars fields we have been considering. The final result for general $N$, presented as a sum formula, is given in eq. . Rather than jumping straight from eq.  to this result, we begin by considering three simpler 