From histograms of time delays between counts in a neutron monitor, one can extract the leader fraction, $L$, of neutron counts that did not follow a previous neutron count in the same counter tube associated with the same cosmic ray shower. The leader fraction is the inverse of the neutron multiplicity and can indicate variations in the cosmic ray spectral index over a certain rigidity range above the local cutoff. Here we calibrate $L$ from the South Pole neutron monitor, with an atmospheric cutoff of $\sim 1$ GV, with respect to a daily spectral index determined using published data from the Alpha Magnetic Spectrometer (AMS-02) aboard the International Space Station for Galactic cosmic ray proton fluxes during 2015-2019. We find that the leader fraction is well correlated with the daily proton spectral index fit over 2.97-16.6 GV as inferred from AMS-02. We estimate the uncertainty in the spectral index as inferred from the leader fraction. This work verifies that the ground-based neutron monitor network can perform precise and continuous monitoring of cosmic ray spectral variations in the long-term future.