Reachability of optimal representations via flips and reductions for matrix multiplication

Determine whether, for the matrix multiplication tensor, there always exists a path using only flip and reduction operations (i.e., excluding splits) from the standard representation to an optimal-rank representation; if such paths exist, derive a bound on the length of the shortest path and characterize its structural properties.

Background

Flip graphs connect tensor representations via flips and reductions, and optionally splits. For polynomial multiplication, the authors prove a bounded-length path from the standard to an optimal representation using flips and reductions only. For matrix multiplication, this remains unresolved.

Clarifying reachability without splits and bounding shortest paths would yield principled search strategies in the flip graph framework, potentially enabling systematic discovery of optimal matrix multiplication algorithms and guiding heuristic design.