cutoff dependence decreases from LO to NLO. While it is conceivable that the residual ${\mathcal{C}}_n$ is sometimes somewhat larger than NDA predicts, NDA should apply “most of the time”, statistically speaking.

Still, a specific regulator form may produce a very small residual cutoff dependence at one order but a significantly larger one at a subsequent order, ${\mathcal{C}}_n(\Lambda_1)-{\mathcal{C}}_n(\Lambda_2)<
{\mathcal{C}}_{n+1}(\Lambda_1)-{\mathcal{C}}_{n+1}(\Lambda_2)$. This may for example occur if the regulator produces only corrections with even powers of $\Lambda$ and the numerics preserves this symmetry at least approximately (e.g. because $\Lambda_1\approx\Lambda_2$, allowing for a perturbative expansion). If this overwhelms the expansion in $Q$, ${\mathcal{O}}$ may indeed systematically become more dependent on $\Lambda$ between some orders, but not between all. Nonetheless, one should not just see some qualitatively improved cutoff dependence with increasing order, but one must see the quantitatively predicted slopes emerge for many orders: they must be $\ge n+1$; see eq. .

[**Constructing a PC by Trial-and-Error**]{} If the cutoff dependence of a given observable does not decrease consistently between subsequent orders, caution may be advisable. For example, $\Lambda$-dependence may increase from one order to the next, but then decrease markedly when another full order with a new LEC is included. This could signal 