  Approximations to the exact density functional for the exchange-correlation
energy of a many-electron ground state can be constructed by satisfying
constraints that are universal, i.e., valid for all electron densities.
Gedanken densities are designed for the purpose of this construction, but need
not be realistic. The uniform electron gas is an old gedanken density. Here, we
propose a spherical two-electron gedanken density in which the dimensionless
density gradient can be an arbitrary positive constant wherever the density is
non-zero. The Lieb-Oxford lower bound on the exchange energy can be satisfied
within a generalized gradient approximation (GGA) by bounding its enhancement
factor or simplest GGA exchange-energy density. This enhancement-factor bound
is well known to be sufficient, but our gedanken density shows that it is also
necessary. The conventional exact exchange-energy density satisfies no such
local bound, but energy densities are not unique, and the simplest GGA
exchange-energy density is not an approximation to it. We further derive a
strongly and optimally tightened bound on the exchange enhancement factor of a
two-electron density, which is satisfied by the local density approximation but
is violated by all published GGA's or meta-GGA's. Finally, some consequences of
the non-uniform density-scaling behavior for the asymptotics of the exchange
enhancement factor of a GGA or meta-GGA are given.
