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We apply a recently-developed low-field technique to inductively measure thecritical pair momentum $p_c$ in thin, underdoped films ofY$_{1-x}$Ca$_{x}$Ba$_{2}$Cu$_{3}$O$_{7-\delta}$ andBi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$ reflecting a wide range of holedoping. We observe that $p_c \propto \hbar/\xi$ scales with $T_c$ and thereforesuperfluid density $n_s(T\rightarrow0)$ in our two-dimensional cuprate films.This relationship was famously predicted by a universal model of the cuprateswith a \textit{doping-independent} superconducting gap, but has not beenobserved by high field measurements of the coherence length $\xi$ due tofield-induced phenomena not included in the theory.

Doping-dependent critical Cooper-pair momentum in thin, underdoped cuprate films