立体矩阵论基础:定义、直和结构与表征能力

摘要 本文是立体矩阵(SM)统一矩阵论系列的首篇,回答纲领的逻辑起点:该对象是什么、能表达什么、如何用它建模。定义层面,给出SM的形式定义体系:N阶立体矩阵为三重张量积空间 V⊗V⊗V 的元素,立体矩阵乘法(SMM)以行-列收缩、公共层保持为指标规则,三个指标各司其职。表征能力层面,核心事实是直和结构定理:SMM 代数同构于N个经典矩阵代数的直和 ⊕ⱼM_N(𝔽)。平面矩阵乘法不是立体矩阵乘法的类比,而是其固定公共层下的真子结构;经典矩阵论(特征值、Jordan型、谱定理)经此同构逐层注入,而SM真正的新内容恰是不被该分解捕获的跨层结构——由此"SM统一平面矩阵"获得严格数学内核。进而论证多元耦合的不可约性:存在不能由任何双元耦合系统表达的三元结构(2×2×2矩阵乘法张量的 𝔽₂ 秩恰为7,含SAT不存在性证书),故三维本体是表达必需而非便利选择。建模层面,给出关系本体论与标准建模流程 M1–M5,并以捕食者-猎物-资源三方生态系统为完整示例。本文同时声明表征能力的诚实边界:规则多元耦合是其主场,纯二元交互网络与统计黑箱不在其承诺之内。

Abstract This is the first paper of the Stereo Matrix (SM) unified matrix theory series, addressing the logical starting point of the programme: what the object is, what it can express, and how to model with it. At the definitional level we give the formal definition system of SM: an order-N stereo matrix is an element of the triple tensor product space V⊗V⊗V, and stereo matrix multiplication (SMM) takes row–column contraction with the common layer preserved as its index rule, each of the three indices playing a distinct role. At the level of representational capacity, the central fact is the direct-sum structure theorem: the SMM algebra is isomorphic to a direct sum of N classical matrix algebras, ⊕ⱼM_N(𝔽). Planar matrix multiplication is not an analogue of stereo matrix multiplication but a proper substructure obtained by fixing the common layer; classical matrix theory (eigenvalues, Jordan form, spectral theorem) is injected layer by layer through this isomorphism, while what is genuinely new in SM is precisely the cross-layer structure not captured by that decomposition — whereby "SM unifies planar matrices" acquires a rigorous mathematical core. We further establish the irreducibility of multi-way coupling: there exist ternary structures inexpressible by any binary-coupling system (the 𝔽₂ rank of the 2×2×2 matrix multiplication tensor is exactly 7, with a SAT non-existence certificate), so the three-dimensional ontology is a necessity of expression rather than a convenience. At the modelling level we give the relational ontology and the standard modelling procedure M1–M5, illustrated end-to-end by a predator–prey–resource three-party ecosystem. We also state the honest boundary of representational capacity: regular multi-way coupling is its home ground, while purely binary interaction networks and statistical black boxes lie outside its promise.

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