Its good to see that you are skeptical, I am as well. Having said that I am also a bit more optimistic, or - more correctly stated - more excited; there is a larger theme surrounding the ideas which most people seem to not have picked up on yet.Even if they did, it would all only be numerology. There is nothing in the paper like an equation of motion, let alone one employing Atiyah's new constant. No alternative method of calculating anything physical is provided.
His idea seems to be: the forces correspond to an algebraic hierarchy (levels I through III), the coupling constants are mathematical constants that appear at the different levels, and exactly how this comes together as physics will be figured out later.
The most positive thing I can say, is that this is a lesson in imagination and in thinking big. The idea that the various couplings will arise as "the noncommutative counterpart of pi" or "the nonassociative counterpart of pi", in the context of a novel algebraic ordering of the fundamental forces, is visionary and systematic. One should hope for and aim for ideas so striking and clear. Nonetheless, this particular idea also seems to be completely wrong.
Quite honestly it's been quite a long while, that I was actually this excited about some mathematical method. The nice thing about Atiyah's equations is that they both tie in quite nicely with some already existing proposals in theoretical physics as well as imply some new things about old theories.
I myself am also trying to reproduce all of his numerics; it may be a wild goose chase but what is there to lose? If it works, this will be the first real progress in theoretical physics in 40 years, and if it doesn't we'll have learned some new potentially useful rapid convergence computational techniques.
I would love to be much more specific about what it is I'm exactly on about, but I don't wanna jump the gun. Needless to say I'd prefer to have the competition for potential new discoveries in theoretical physics based on this remain at a bare minimum as well