  We consider the techniques behind the current best algorithms for matrix
multiplication. Our results are threefold.
  (1) We provide a unifying framework, showing that all known matrix
multiplication running times since 1986 can be achieved from a single very
natural tensor - the structural tensor $T_q$ of addition modulo an integer $q$.
  (2) We show that if one applies a generalization of the known techniques
(arbitrary zeroing out of tensor powers to obtain independent matrix products
in order to use the asymptotic sum inequality of Sch\"{o}nhage) to an arbitrary
monomial degeneration of $T_q$, then there is an explicit lower bound,
depending on $q$, on the bound on the matrix multiplication exponent $\omega$
that one can achieve. We also show upper bounds on the value $\alpha$ that one
can achieve, where $\alpha$ is such that $n\times n^\alpha \times n$ matrix
multiplication can be computed in $n^{2+o(1)}$ time.
  (3) We show that our lower bound on $\omega$ approaches $2$ as $q$ goes to
infinity. This suggests a promising approach to improving the bound on
$\omega$: for variable $q$, find a monomial degeneration of $T_q$ which, using
the known techniques, produces an upper bound on $\omega$ as a function of $q$.
Then, take $q$ to infinity. It is not ruled out, and hence possible, that one
can obtain $\omega=2$ in this way.
