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

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, η_friction, ε_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]. 

 

Modeling high-dimensional data and AI embeddings as pushforward measures on a metric-measure space (X, d, μ) equipped with Bakry–Émery Ricci curvature, Ollivier–Ricci discrete curvature, and 16D Sedenion–Cartan torsion T_[λμν], 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) < ε ≰ d_geo(x,y) < δ) driven by small effective curvature and non-associative torsion; (3) Thin-Shell Concentration around radius √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(√(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.