立体矩阵在学科地图上的位置:与张量方法与分解算法的关系

摘要 统一矩阵论不是孤立构造:它与至少六个既有学科板块相邻。本文逐一建立关系并定级。其一,与主流张量分解(CPD/Tucker)是对照关系——它们把三阶张量当作分解对象,SM把三阶对象当作乘法的承载体,公共层指标永不收缩是分水岭。其二,与 Kilmer 学派的 t-积是分工关系——t-积沿第三维循环卷积、本质上是块循环矩阵乘法的傅里叶对角化;SMM 保留公共层不收缩,由直和结构定理刻画,本文附录给出可运行的数值对照。其三,与 AlphaTensor/flip-cpd 等分解算法研究是分工关系——它们回答如何更快搜到短分解,本文回答分解空间的全局地貌是什么、什么可发证。其四,与 Montgomery 形状空间、Barbour 形状动力学是对译关系——三体构型对相似群取商正是形状空间,商化周期判据是其SM影像。其五,与规范场论按 Wu–Yang 字典的精神处理为纲领级对译(纤维=公共层、截面=切片选取、holonomy=绕界面的相位),仅标【约定】。其六,与 SOC/网络科学是现象类比——爆发节律与团簇景观形态相似而机制不同,如实划界。最后给出三个核心问题的阶段性解答总表与开放问题。

Abstract Abstract: Unified matrix theory is not an isolated construction: it is adjacent to at least six existing disciplinary blocks. This paper establishes and grades each relation. First, with mainstream tensor decomposition (CPD/Tucker) the relation is one of contrast — they treat third-order tensors as objects of decomposition, SM treats a third-order object as the carrier of multiplication, and the never-contracted common-layer index is the watershed. Second, with the Kilmer school's t-product the relation is one of division of labour — the t-product convolves along the third dimension and is essentially Fourier diagonalization of block-circulant matrix multiplication, whereas SMM preserves the common layer uncontracted and is characterized by the direct-sum structure theorem; a runnable numerical comparison is given in the appendix. Third, with decomposition-algorithm research such as AlphaTensor/flip-cpd the relation is one of division of labour — they ask how to search faster for short decompositions, this paper asks what the global landscape of the decomposition space is and what can be certified. Fourth, with Montgomery's shape space and Barbour's shape dynamics the relation is one of translation — the quotient of three-body configurations by the similarity group is precisely shape space, and the quotient-period criterion is its SM image. Fifth, with gauge field theory the relation is handled in the spirit of the Wu–Yang dictionary as a programme-level translation (fibre = common layer, section = slice selection, holonomy = phase around an interface), graded as a convention only. Sixth, with SOC/network science the relation is a phenomenological analogy — bursty rhythm and clustered landscape are similar in form but different in mechanism, and the boundary is drawn honestly. A summary table of provisional answers to three core questions and open problems concludes the paper.

本记录含中文版与英文版两个文件(Bilingual: Chinese + English DOCX)。