  Recently, it was discovered that certain non-Hermitian systems can exhibit
qualitative different properties at different system sizes, such as being
gapless at small sizes and having topological edge modes at large sizes $L$.
This dramatic system size sensitivity is known as the critical non-Hermitian
skin effect (cNHSE), and occurs due to the competition between two or more
non-Hermitian pumping channels. In this work, we rigorously develop the notion
of a size-dependent generalized Brillouin zone (GBZ) in a general
multi-component cNHSE model ansatz, and found that the GBZ exhibits a universal
$a+b^{1/(L+1)}$ scaling behavior. In particular, we provided analytical
estimates of the scaling rate $b$ in terms of model parameters, and
demonstrated their good empirical fit with two paradigmatic models, the coupled
Hatano-Nelson model with offset, and the topologically coupled chain model with
offset. We also provided analytic result for the critical size $L_c$, below
which cNHSE scaling is frozen. The cNHSE represents the result of juxtaposing
different channels for bulk-boundary correspondence breaking, and can be
readily demonstrated in non-Hermitian metamaterials and circuit arrays.
