We study N=4 SYM at non-integer number of colours. By varying N
we can continuously follow states all the way from N=β where
integrability reigns to finite N where quantum gravity effects dominate. As
an application we consider classically 1/16 BPS states. Quantum mechanically,
these states are generically non-supersymmetric but some special states -- at
special values of N -- become super-symmetric at the quantum level as well.
They are the so-called quantum black hole states studied recently using
cohomology. We write down the form of the lightest BH state at N=2 -- and
follow it in N, both at weak coupling and -- more speculatively -- at strong
coupling as well. At weak coupling this state has protected dimension
Ξ=19/2 at N=2 and becomes a triple trace made out of Konishi and two
light BPS operators at infinite N with Ξ=19/2+12Ξ»+β¦. At
strong coupling we suspect it becomes a quadruple trace with dimension Ξβ19/2+integer.Comment: 7+8 pages, 20 footnote