  We study the algorithmic problem of multiplying large matrices that are
rectangular. We prove that the method that has been used to construct the
fastest algorithms for rectangular matrix multiplication cannot give optimal
algorithms. In fact, we prove a precise numerical barrier for this method. Our
barrier improves the previously known barriers, both in the numerical sense, as
well as in its generality. We prove our result using the asymptotic spectrum of
tensors. More precisely, we crucially make use of two families of real tensor
parameters with special algebraic properties: the quantum functionals and the
support functionals. In particular, we prove that any lower bound on the dual
exponent of matrix multiplication $\alpha$ via the big Coppersmith-Winograd
tensors cannot exceed 0.625.
