necessarily an integer, as Table \[tab:pc\] shows, and the first omitted order is not always $Q^{n+1}$, but more generally $Q^{n+\alpha}$, $\mathrm{Re}[\alpha]>0$. To replace $n+1\to n+\alpha$ in eqs. , , – and indeed throughout – is straightforward. In [EFT($\slashed{\pi}$)]{}, the slope-fit in eq.  endorses that the $3N$ PC proceeds in integer steps. Including non-analytic dependencies of the residuals on $k$ or ${p_\text{typ}}$ is also straightforward. For the remainder of the presentation, all such replacements are implied, but we stick to the integer case for convenience.

[**Assumptions of the Expansion**]{} The assumptions on the residual ${\mathcal{C}}_n$ are endorsed if order $n$ and breakdown scale ${\overline{\Lambda}_\text{EFT}}$ follow indeed the functional form of eq.  or its variant . Naïve Dimensional Analysis (NDA) sets the magnitude of ${\mathcal{C}}_n$ to the scale of its running [@NDA; @NDA2]. Its cutoff-dependence and other effects are eventually absorbed into higher-order LECs (see also below).

We can actually be somewhat more specific about the condition that the variation of the residual ${\mathcal{C}}_n$ variation with respect to $\Lambda$ should be larger than that for other parameters. Since $k,{p_\text{typ}}\ll\Lambda_1,\Lambda_2$, the dimensionless ratio on the left-hand side of eq.  can be expanded as $$\label{eq:residualexpansion}
  \frac{{\mathcal{C}}_n(\Lambda_1;k,{p_\text{typ}},{\overline{\Lambda}_\text{EFT}})-
    {\mathcal{C}}_n(\Lambda_2;k,{p_\text{typ}},{\overline{\Lambda}_\text{EFT}})} 
  {{\mathcal{C}}_n(\Lambda_1;k,{p_\text{typ}},{\overline{\Lambda}_\text{EFT}})}= c_0(\Lambda_1,\Lambda_2;{\overline{\Lambda}_\text{EFT}})+
  c_1(\Lambda_1,\Lambda_2;{\overline{\Lambda}_\text{EFT}})\;\frac{k,{p_\text{typ}}}{\Lambda_1,\Lambda_2}+\dots$$ If the first term 