We present a quantitative analysis of superfluidity and superconductivity in dense matter from observations of isolated neutron stars in the context of the minimal cooling model. Our new approach produces the best fit neutron triplet superfluid critical temperature, the best fit proton singlet superconducting critical temperature, and their associated statistical uncertainties. We find that the neutron triplet critical temperature is likely $2.09^{+4.37}_{-1.41} times 10^{8}$ K and that the proton singlet critical temperature is $7.59^{+2.48}_{-5.81} times 10^{9}$ K. However, we also show that this result only holds if the Vela neutron star is not included in the data set. If Vela is included, the gaps increase significantly to attempt to reproduce Velas lower temperature given its young age. Further including neutron stars believed to have carbon atmospheres increases the neutron critical temperature and decreases the proton critical temperature. Our method demonstrates that continued observations of isolated neutron stars can quantitatively constrain the nature of superfluidity in dense matter.
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