  We examine key aspects of the theory of the Bardeen-Cooper-Schrieffer (BCS)
to Bose-Einstein condensation (BEC) crossover, focusing on the temperature
dependence of the chemical potential, $\mu$. We identify an accurate method of
determining the change of $\mu$ in the cuprate high temperature superconductors
from angle-resolved-photoemission data (along the "nodal" direction), and show
that $\mu$ varies by less than a few percent of the Fermi energy over a range
of temperatures from far below to several times above the superconducting
transition temperature, $T_c$. This shows, unambiguously, that not only are
these materials always on the BCS side of the crossover (which is a phase
transition in the $d-wave case), but are nowhere near the point of the
crossover (where the chemical potential approaches the band bottom).
