A catalog of fast matrix multiplication algorithms with frontier-closure search

Abstract

The 2022--2026 burst of activity in small-format matrix multiplication (AlphaTensor 2022, AlphaEvolve 2025, Schwartz--Zwecher 2025) has produced striking individual results but scattered them across different fields, attribution conventions, and serialisation formats. A complementary line of work -- Perminov's open-source flip-graph framework~\cite{perminov2026fast,perminov2025fast} -- instead drives existing construction methods, notably flip-graph and \emph{meta-flip-graph} search, at scale across large format spaces, discovering many new low-rank schemes (including ternary-integer ones) that further enrich the landscape this catalog must unify. We present a unified, machine-checkable catalog covering shapes up to \nmpshape{32}{32}{32} over \Rationals, \Integers, \Reals, \Complex, and \Ftwo, with a separate axis for commutative algorithms (Waksman 1970, Makarov 1986, Rosowski 2019). Derivation over this catalog is performed by a \emph{frontier-closure search} that recombines catalog entries by axis-flip, Kronecker, axis concatenation, serendipitous products, recombination-with-allocation (with optional output peeling and pair fusion), and downward projection. A central methodological point is the \emph{non-overlap property}: our recombination does not, and cannot, rediscover the shared bilinear products that hand-crafted constructions (Strassen, Laderman, Smirnov, AlphaTensor) are built around. This draws a clean line between the ``find a cleverer bilinear core'' and ``compose known cores'' axes of progress, and resolves several attribution puzzles in the literature. We refresh the DIS09 comparison tables, split per field and with a commutative column, and provide the tooling to regenerate them automatically as the catalog evolves.