This is the title of an interesting paper by Zaliapin and Ghil that appeared some time ago on the
Arxiv and more recently in peer-reviewed form in
NPG. It's presented as a criticism of Roe and Baker, which has already been
debunked enough, so after a quick glance I didn't pay too much attention to it at first. I also know the second author to be extremely clever, so I was worried that it might be really technical. On a second, slower, reading, however, it's actually quite straightforward and very interesting. It also seems rather harsh on R&B, because the criticism applies to a whole host of similar work.
Z&G start with the basic premise that R&B - and indeed all work of this nature - use, that there's a functional relationship between radiation R (at the top of the atmosphere) and the surface temperature T, which we can write as R=R(T,a(T)) where the notation indicates that a(T) is the "feedback" term, that is to say, if we replace a(T) with zero then the function returns the zero-feedback relationship.
We can then perform a Taylor series expansion to investigate how the radiation balance changes with temperature:
DR = dR/dT*DT + dR/da*da/dT*DT + O(DT
2)
(read the real paper for more elegant typesetting)
By writing 1/L0 = dR/dT (L0 is the zero-feedback sensitivity) and defining f=-L0*dR/da*da/dT we arrive at the familiar expression
DR = (1-f)/L0 * DT + O(DT
2)
Now what everyone does at this point is to drop the last term and use the linear approximation, which can be re-arranged to give
DT = L0/(1-f) * DF
exhibiting the well-known singularity for a feedback of f=1.
What Zaliapin and Ghil point out is really startlingly simple and IMO elegant. They merely observe that if f is close to one, the linear truncation was not justified because the quadratic term is now large enough to matter! Once it is included, the singularity at f=1 goes away, as their Fig 2 shows:

(The feedback factor f cannot be larger than one or the initial equilibrium is unstable, even with the nonlinear term, hence the upper bound on the x-axis is sound.)
I'm not yet sure how much this really matters. We can still get a high sensitivity so long as the nonlinearity is small. AIUI most models do show a fairly, but not perfectly, linear response over quite a range of forcing and temperature, and the existence of complex climate models with sensitivities above 6C implies that such high values are at least not a mathematical impossibility. It may, however, make it harder to justify the sort of pathological "long tail" arguments beloved by some. Of course I've argued against them on a number of grounds already - not least of which is that, from a policy perspective, we are on really shaky ground if all the calls to action have to be based on highly improbable events that we are pretty confident won't happen irrespective of what mitigation we do or do not attempt. In any case, the maths is interesting in its own right.