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.

翻译：2022年至2026年间小规模矩阵乘法（AlphaTensor 2022、AlphaEvolve 2025、Schwartz-Zwecher 2025）的爆发式研究活动产出了引人注目的个别结果，但这些结果分散在不同领域、归因惯例和序列化格式中。另一条互补的研究路线——Perminov开源的翻转图框架~\cite{perminov2026fast,perminov2025fast}——则在大规模格式空间中驱动现有构造方法（尤其是翻转图和元翻转图搜索）规模化运行，发现了众多新的低秩方案（包括三元整数方案），进一步丰富了本目录必须统一的研究图景。我们提出一个统一且可机器校验的目录，涵盖\textbf{规模不超过}\nmpshape{32}{32}{32}的\textbf{有理数域}、\textbf{整数环}、\textbf{实数域}、\textbf{复数域}和\textbf{二元域}上的情形，并单独设置交换算法分类轴（Waksman 1970、Makarov 1986、Rosowski 2019）。该目录的推演通过\textbf{前沿闭包搜索}实现：通过轴翻转、Kronecker积、轴拼接、意外乘积、带分配的重新组合（可选输出剥离与配对融合）以及向下投影对目录条目进行重组。一个核心方法论要点是\textbf{非重叠性质}：我们的重组不会且无法重现手工构造（Strassen、Laderman、Smirnov、AlphaTensor）所依赖的共享双线性乘积。这在"寻找更巧妙的双线性核心"与"组合已知核心"这两条进展轴之间划清了界限，并解决了文献中若干归因难题。我们更新了DIS09比较表，按领域拆分并增设交换数列，同时提供工具支持，使该表能随目录演化自动再生。