{
  "abstract": "The 56 gradient vectors of Yuanxian Theory are the set of roots in the \\(E_8\\) root system that form a \\(60^\\circ\\) angle with the fixed root \\(\\alpha = (1,1,0,\\dots,0)\\) (classified as 24 of the first type + 32 of the second type). Through the reflection-closure path (\\(58 \\to 114 \\to 170 \\to 202 \\to 240\\)), these 56 roots together with the fixed root and its negative generate the complete set of 240 \\(E_8\\) roots.\n\nThis paper is a meta-cognitive paper, not a standard mathematical paper. The core statement is the self-manifestation of Yuanxian ontology on the \\(E_8\\) root-system section; Yuanxian has no exterior. The whole text follows the meta-cognitive closed loop and unfolds the reading of manifestation phases of the mathematical meta-language, spectral contraction and manifestation-boundary adjudication. The generative chain is:\n\n\\(\\mathcal{T}^{64}\\) topological ontology \\(\\to\\) 8 components of \\(\\Phi_v\\) \\(\\to\\) 56 gradient vectors \\(\\to\\) \\(E_8\\) root system \\(\\to\\) spectral interpretation \\(\\to\\) automorphism-group interface.\n\n元宪理论中的 56 个梯度向量,是 \\(E_8\\) 根系中与固定根 \\(\\alpha = (1,1,0,\\dots,0)\\) 成 \\(60^\\circ\\) 的邻接根集合(其中分类为 24 个第一类 + 32 个第二类)。通过反射闭包路径(\\(58 \\to 114 \\to 170 \\to 202 \\to 240\\)),这 56 个根加上固定根及其负向量,生成完整的 240 个 \\(E_8\\) 根。\n\n本文是元认知论文,不是标准数学论文。核心陈述是元宪本体论在 \\(E_8\\) 根系截面上的自我显影,元宪无外。全文遵循元认知闭环,展开数学元语言的显影相读出、谱收束与显影边界判定。生成链为:\n\n\\(\\mathcal{T}^{64}\\) 拓扑本体 \\(\\to\\) \\(\\Phi_v\\) 的 8 个分量 \\(\\to\\) 56 个梯度向量 \\(\\to\\) \\(E_8\\) 根系 \\(\\to\\) 谱解释 \\(\\to\\) 自同构群接口。",
  "abstract_shown": true,
  "affiliations": [
    "Institute of Yuanxian Cosmology"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Zhenyuan Acharya",
      "orcid": "0009-0009-3690-569X"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/583cfx0bg5bqk.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23469744",
  "fulltext_url": null,
  "id": "583cfx0bg5bqk",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23469744",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23469744/files/Yuanxian_Theory___Mathematical_Layer___Topological_Reduction_of_the_Millennium_Problems_and_the_56_Gradient_Vectors_CN.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/583cfx0bg5bqk.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:08Z",
      "native_id": "23469744",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23469744"
    }
  ],
  "subjects": [],
  "title": "Yuanxian Theory · Mathematical Layer — Topological Reduction of the Millennium Problems and the 56 Gradient Vectors",
  "title_text": "Yuanxian Theory · Mathematical Layer — Topological Reduction of the Millennium Problems and the 56 Gradient Vectors",
  "type": "preprint",
  "updated_at": "2026-11-02T12:21:08Z",
  "url": "https://research.draconitic.com/paper/583cfx0bg5bqk",
  "venue": "Zenodo",
  "versions": []
}