Strassen founded the theory of the asymptotic spectrum of tensors to study the complexity of matrix multiplication. A central challenge in this theory is to explicitly construct new spectral points. In Crelle 1991, Strassen proposed the upper support functionals $ζ^θ$ as candidate spectral points, where $θ$ ranges over a triangle $Θ$. Recent progress, involving tools and ideas from quantum information theory (Christandl-Vrana-Zuiddam, STOC 2018, JAMS 2021) and convex optimization (Hirai, 2025), culminated in the proof that the upper support functionals are indeed spectral points over the complex numbers (Sakabe-Doğan-Walter, 2026). In this paper, we give an even clearer picture of the situation for support functionals when $θ$ lies along the edges of the triangle. We show that not only are these functionals spectral points, but that they are uniquely determined as spectral points by their behavior on matrix multiplication tensors. As our methods are algebraic, as a corollary this establishes for the first time the existence of nontrivial spectral points over arbitrary fields. As part of our argument, we show a close connection between the edge support functionals and Harder-Narasimhan filtrations from quiver representation theory. We thus show, using recent work in algorithmic invariant theory, that these support functionals can be computed in deterministic polynomial time. Other ingredients of our proof include a new criterion for abstractly characterizing asymptotic tensor ranks by spectral points, and a characterization of the edge support functionals in terms of matrix multiplication capacity. As another application of these tools, we prove the existence of spectral points for higher-mode tensors beyond those currently known.

翻译：斯特拉森建立了张量渐近谱理论，以研究矩阵乘法的复杂性。该理论的一个核心挑战是显式构造新的谱点。在Crelle 1991中，斯特拉森提出了上支撑泛函$ζ^θ$作为候选谱点，其中$θ$取值范围为三角形$Θ$。近期，借助量子信息理论（Christandl-Vrana-Zuiddam, STOC 2018, JAMS 2021）和凸优化（Hirai, 2025）的工具与思想，取得了进展，最终证明了上支撑泛函确实是复数域上的谱点（Sakabe-Doğan-Walter, 2026）。本文中，我们对$θ$位于三角形边界时的支撑泛函情况给出了更清晰的描述。我们证明这些泛函不仅是谱点，而且由它们在矩阵乘法张量上的行为唯一确定为谱点。由于我们的方法是代数的，作为推论，这首次确立了任意域上非平凡谱点的存在性。作为论证的一部分，我们展示了边界支撑泛函与箭图表示论中Harder-Narasimhan滤链之间的密切联系。因此，利用算法不变理论的最新工作，我们证明这些支撑泛函可以在确定性多项式时间内计算。证明的其他要素包括：用谱点抽象刻画渐近张量秩的新准则，以及用矩阵乘法容量刻画边界支撑泛函。作为这些工具的另一应用，我们证明了当前已知范围之外的高阶张量的谱点存在性。