{
  "abstract": "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)。",
  "abstract_shown": true,
  "affiliations": [
    "W3 Lab"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Liang Wu",
      "orcid": "0009-0006-9694-5739"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9573xmm6nd31c.bib",
  "cited_by": 17,
  "doi": "10.5281/zenodo.23430222",
  "fulltext_url": null,
  "id": "9573xmm6nd31c",
  "issns": [],
  "landing_url": "https://doi.org/10.5281/zenodo.23430222",
  "language": "cmn",
  "licence": "CC-BY-NC",
  "licence_label": "CC BY-NC",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by-nc/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2605|Tensor decomposition and applications"
    ]
  ],
  "openalex": "W7222809190",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": "European Organization for Nuclear Research",
  "ris": "https://research.draconitic.com/paper/9573xmm6nd31c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:17Z",
      "native_id": "W7222809190",
      "source": "openalex",
      "url": "https://doi.org/10.5281/zenodo.23430222"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2605|Tensor decomposition and applications",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "立体矩阵论基础:定义、直和结构与表征能力",
  "title_text": "立体矩阵论基础:定义、直和结构与表征能力",
  "type": "preprint",
  "updated_at": "2026-11-08T13:20:35Z",
  "url": "https://research.draconitic.com/paper/9573xmm6nd31c",
  "venue": "Zenodo (CERN European Organization for Nuclear Research)",
  "versions": []
}