  We advance the Cohn-Umans framework for developing fast matrix multiplication
algorithms. We introduce, analyze, and search for a new subclass of strong
uniquely solvable puzzles (SUSP), which we call simplifiable SUSPs. We show
that these puzzles are efficiently verifiable, which remains an open question
for general SUSPs. We also show that individual simplifiable SUSPs can achieve
the same strength of bounds on the matrix multiplication exponent $\omega$ that
infinite families of SUSPs can. We report on the construction, by computer
search, of larger SUSPs than previously known for small width. This, combined
with our tighter analysis, strengthens the upper bound on the matrix
multiplication exponent from $2.66$ to $2.505$ obtainable via this
computational approach, and nears the results of the handcrafted constructions
of Cohn et al.
