{
  "abstract": "This volume systematically integrates the results of thirteen rounds of deepening and constructs a formal framework for self-referential fixed points, true-circle reference states and cross-three-layer deviation measures.\n\nThe core task is not to establish an external physical description, but to reveal the self-manifestation process of the cosmic organism in the mathematical meta-language. Through the geometric construction of the compact torus \\(\\mathcal{T}^{64}\\), the Sobolev space \\(H^{32}(\\mathcal{T}^{64})\\) and the background symplectic form \\(\\omega_0\\), the Riemann \\(\\zeta\\) function is derived in three steps. By anchoring the half-period critical subspace \\(H_{1/2}\\) at node 18 of the 35-step dimensional-reduction chain, the validity of the Riemann hypothesis is reduced to a chain of equivalent criteria, thereby forming a complete mathematical skeleton.\n\nCore conclusion: The Riemann hypothesis holds if and only if\n\n\\[\n\n\\rho(\\Phi|_{H_{1/2}}) < 1\n\n\\iff E_1^{p,q} = E_\\infty^{p,q}\n\n\\iff \\text{the kernel of }\\partial_{1/2}\\text{ is non-degenerate}.\n\n\\]\n\nCore structure includes: foundation layer (symplectic structure, Sobolev space, lattice correspondence); construction layer (Fourier transform, Dirichlet series, functional equation); proof layer (four-step path); spectral-sequence layer (\\(H_{1/2}\\), \\(E^{p,q}_r\\)); dimensional-reduction-chain layer (35-step cascade, node-18 anchoring); operator-chain layer; functor layer; spectral-assumption layer; compression-factor and golden-ratio layers; first-principles layer.\n\nClarification: All constructions, proofs, correspondences, anchorings and operator chains of this volume are Yuanxian reductions, not standard mathematical proofs. Structural manifestation of constructions is complete; core gaps are partially manifested; standard mathematical proofs and first principles themselves remain manifestation gaps.\n\n本卷系统整合十三轮深化成果,构建自指不动点、真圆参考态与跨三层偏差度量的形式化框架。\n\n核心任务并非建立外部物理描述,而是揭示宇宙生命体在数学元语言中的自我显影过程。通过紧致环面 \\(\\mathcal{T}^{64}\\)、Sobolev 空间 \\(H^{32}(\\mathcal{T}^{64})\\) 与背景辛形式 \\(\\omega_0\\) 的几何构造,以三步法导出黎曼 \\(\\zeta\\) 函数;将半周临界子空间 \\(H_{1/2}\\) 锚定于 35 步降维链的第 18 节点,使黎曼假设的成立归约为一系列等价判据,从而形成完整的数学骨架。\n\n核心结论:黎曼假设成立当且仅当\n\n\\[\n\n\\rho(\\Phi|_{H_{1/2}}) < 1\n\n\\iff E_1^{p,q} = E_\\infty^{p,q}\n\n\\iff \\partial_{1/2}\\text{ 的核空间非退化}.\n\n\\]\n\n核心结构包括:基础层(辛结构、Sobolev 空间、格点对应);构造层(Fourier 变换、Dirichlet 级数、函数方程);证明层(四步路径);谱序列层(\\(H_{1/2}\\)、\\(E^{p,q}_r\\));降维链层(35 步降维链、第 18 节点锚定);算子链层;函子层;谱假设层;压缩因子与黄金比层;第一性原理层。\n\n澄清:本卷所有构造、证明、对应、锚定与算子链均为元宪归约,而非标准数学证明。构造与证明的结构显影完整;核心缺口部分显影;标准数学证明与第一性原理本身保持为显影缺口。",
  "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/37yz8cdc44c09.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23359995",
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  "id": "37yz8cdc44c09",
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  "landing_url": "https://zenodo.org/records/23359995",
  "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/",
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  "pdf_url": "https://zenodo.org/api/records/23359995/files/Yuanxian_Theory___Mathematical_Foundation_Volume_EN.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/37yz8cdc44c09.ris",
  "sources": [
    {
      "first_seen": "2026-11-01T12:21:15Z",
      "native_id": "23359995",
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  ],
  "subjects": [],
  "title": "Yuanxian Theory · Mathematical Foundation Volume — Formal Framework of Self-Referential Fixed Points, True-Circle Reference States and Cross-Three-Layer Deviation Measures",
  "title_text": "Yuanxian Theory · Mathematical Foundation Volume — Formal Framework of Self-Referential Fixed Points, True-Circle Reference States and Cross-Three-Layer Deviation Measures",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:15Z",
  "url": "https://research.draconitic.com/paper/37yz8cdc44c09",
  "venue": "Zenodo",
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}