1) Do not get me wrong. I like the work by Andrew and Virginia.
Both are great pieces of hard work.
But if you know the work by Coppersmith and Winograd (and from
what I read in other comments/posts, not many people do), then
you know the limitations of their methods before reading their papers. I do not have a formal lower bound of what you can get
out of the outer structure of the Coppersmith and Winograd
tensor, but from what one typically observes, the improvement
shrinks geometrically with the power you take.

2) One possible reason why you do
not get anything from the third power is that it is the product
of the second and the first power. The second power is
superior to the first power. Note that taking the second,
fourth or whatever power is done for the sake of analysis.
In the construction, you take a huge power of the tensor.
And then it is better to think of it as a product of
second powers of the tensor than of a product of
second powers as well as first powers. (Again, I cannot
prove this.)

3) Ian Gordon, the internal examiner, and myself told
Andrew after his viva to put a preprint online as soon
as possible. Read the update in the post why he did not.

4) “The calculations in Stother’s paper are correct” in my
comment means that they are not wrong.
Some comments before raised the question whether there
was a serious flaw in Stother’s work. It is not.

I also agree with Cohn: if there is no serious flaw in Stothers’ work, we should credit Stothers’ work as the first that breaks CW barrier. The Williams’s work could be considered as an independent great result and vast improvement on Stothers’ work.