  We use numerical methods to obtain moduli-dependent Calabi-Yau metrics and
from them the moduli-dependent massive tower of Kaluza-Klein states for the
one-parameter family of quintic Calabi-Yau manifolds. We then compute geodesic
distances in their K\"ahler and complex structure moduli space using exact
expressions from mirror symmetry, approximate expressions, and numerical
methods and compare the results. Finally, we fit the moduli-dependence of the
massive spectrum to the geodesic distance to obtain the rate at which states
become exponentially light. The result is indeed of order one, as suggested by
the swampland distance conjecture. We also observe level-crossing in the
eigenvalue spectrum and find that states in small irreducible representations
of the symmetry group tend to become lighter than states in larger irreducible
representations.
