  In earlier work, we (KI and BW) gave a two line "almost proof" (for
supersymmetric RG flows) of the weakest form of the conjectured 4d a-theorem,
that a_{IR}<a_{UV}, using our result that the exact superconformal R-symmetry
of 4d SCFTs maximizes a=3Tr R^3-Tr R. The proof was incomplete because of two
identified loopholes: theories with accidental symmetries, and the fact that
it's only a local maximum of \it{a}. Here we discuss and extend a proposal of
Kutasov (which helps close the latter loophole) in which a-maximization is
generalized away from the endpoints of the RG flow, with Lagrange multipliers
that are conjectured to be identified with the running coupling constants.
a-maximization then yields a monotonically decreasing "a-function" along the RG
flow to the IR. As we discuss, this proposal in fact suggests the strongest
version of the a-theorem: that 4d RG flows are gradient flows of an a-function,
with positive definite metric. In the perturbative limit, the RG flow metric
thus obtained is shown to agree precisely with that found by very different
computations by Osborn and collaborators. As examples, we discuss a new class
of 4d SCFTs, along with their dual descriptions and IR phases, obtained from
SQCD by coupling some of the flavors to added singlets.
