  In this paper we develop further the linear functional method of deriving
lower bounds on the boundary entropy of conformal boundary conditions in 1+1
dimensional conformal field theories (CFTs). We show here how to use detailed
knowledge of the bulk CFT spectrum. Applying the method to the Monster CFT with
c=\bar c=24 we derive a lower bound s > - 3.02 x 10^{-19} on the boundary
entropy s=ln g, and find compelling evidence that the optimal bound is s>= 0.
We show that all g=1 branes must have the same low-lying boundary spectrum,
which matches the spectrum of the known g=1 branes, suggesting that the known
examples comprise all possible g=1 branes, and also suggesting that the bound
s>= 0 holds not just for critical boundary conditions but for all boundary
conditions in the Monster CFT. The same analysis applied to a second bulk CFT
-- a certain c=2 Gaussian model -- yields a less strict bound, suggesting that
the precise linear functional bound on s for the Monster CFT is exceptional.
