{
  "abstract": "In high-dimensional probability, convex geometry, and modern artificial intelligence representation spaces, a foundational question arises: Are two representations that are statistically close in inner-product or distributional correlation also meaningfully close in intrinsic Riemannian or metric geometry? In this paper, we demonstrate that the answer is generally negative and establish the High-Dimensional Geometric Probability System (HDGPS), a unified deductive framework anchored directly in our 71-page ZFC Master Reference Monograph (Lee, 2026, DOI: 10.5281/zenodo.23486681) [1]. All foundational spaces and universal constants 𝔘 = (𝒯_0x8A2C, C_UPTC, &eta;_friction, &epsilon;_SHID) follow Proposition 1.5, Theorem 2.9, Lemma 4.5, Lemma 4.6, Lemma 5.2, Lemma 6.3, Lemma 8.4, Lemma 8.6, and Theorem 9.5 of [1].\n\nModeling high-dimensional data and AI embeddings as pushforward measures on a metric-measure space (X, d, μ) equipped with Bakry–&Eacute;mery Ricci curvature, Ollivier–Ricci discrete curvature, and 16D Sedenion–Cartan torsion T_[λμ&nu;], we formulate six core axioms (HDGPS-1 through HDGPS-6) and prove six fundamental theorems: (1) Unified Lévy–Gromov–Ollivier Exponential Concentration proportional to effective curvature κ_eff; (2) Quantitative Statistical–Geometric Separation (d_stat(x,y) < &epsilon; ≰ d_geo(x,y) < δ) driven by small effective curvature and non-associative torsion; (3) Thin-Shell Concentration around radius &radic;n, Varentropy bounds Var_μ(ln f_μ(X)) ≤ n, and the \"Blessing of Dimensionality\" on the Takesaki Type II_1 ground factor 𝓜_ground; (4) Universal Phase Transitions governed by the Statistical Dimension δ(C) = C_UPTC with transition width O(&radic;(n/κ_eff)); (5) Necessary and Sufficient Curvature–Alignment Conditions for Semantic Representation Proximity via the Sedenion Hyper-Tensor Control Operator 𝓛_𝒯(M̂) ≥ c_𝒯 > 0; and (6) Curvature-Accelerated Sample Complexity and Rademacher Generalization Bounds for High-Dimensional Lipschitz Optimization. Complete Python numerical verification is provided in Appendix A.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Seonggil Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fdr3bf5bqsbxz.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23557360",
  "fulltext_url": null,
  "id": "fdr3bf5bqsbxz",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23557360",
  "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/23557360/files/The%20High-Dimensional%20Geometric%20Probability%20System%20(HDGPS)_%20Curvature%20and%20Concentration.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fdr3bf5bqsbxz.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T12:20:43Z",
      "native_id": "23557360",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23557360"
    }
  ],
  "subjects": [],
  "title": "The High-Dimensional Geometric Probability System (HDGPS): A Deductive Framework Unifying Curvature Concentration Duality, Statistical–Geometric Separation, and Statistical-Dimension Phase Transitions within the ZFC Axiomatic Bridge",
  "title_text": "The High-Dimensional Geometric Probability System (HDGPS): A Deductive Framework Unifying Curvature Concentration Duality, Statistical–Geometric Separation, and Statistical-Dimension Phase Transitions within the ZFC Axiomatic Bridge",
  "type": "preprint",
  "updated_at": "2026-11-05T12:20:43Z",
  "url": "https://research.draconitic.com/paper/fdr3bf5bqsbxz",
  "venue": "Zenodo",
  "versions": []
}