Three-Dimensional Numerical Phase Geometry System: Theory of Geometric Stability for Non-Trivial Zeros — Applications and Limitations in the Study of the Riemann Hypothesis 三维数相位几何体系:非平凡零点分布的几何稳定性理论及其在黎曼假设研究中的应用与局限

Abstract This paper proposes and systematically develops a Three-Dimensional Numerical Phase Geometry System (3DNPG). Its core objective is to understand the distribution law of non-trivial zeros of the Riemann ζ function from the perspective of geometric stability. The core idea of this system is to map the behavior of the ζ function within the critical band into a three-dimensional phase space, and analyze geometric objects including timelike/spacelike structures, isotropic boundaries, contour integrals, and many-body couplings to reveal the underlying reason why the critical line σ=1/2 serves as the "most stable direction". Through twenty series of systematic experiments from Experiment A to Experiment T, we obtain the following results: Algebraic Foundations: Commutativity of multiplication, exact satisfaction of Leibniz rule, approximate associativity with error O(ε) when the δ component is small; Control of Non-Associative Errors: It is proven that weak δ modulation does not alter the topological class of zeros; Soft Transition of Isotropic Boundaries: The isotropic boundary is found to be a soft transition region rather than a hard singularity; 48% of contour points satisfy norm<0 yet the integral does not diverge; Local Stability: The proportion of timelike points equals zero at σ=0.5, while the spacelike proportion reaches 0.798 at σ=1.0; the region near ρ₁ is fully timelike; Contour Geometry and Residue Theory: The integral kernel takes an approximate gradient form; non-zero curl originates from the non-analyticity of finitely truncated Euler products; Many-Body Coupling: A universal scaling law is discovered; Analytic Continuation: The integral functional equation holds with error less than 3%; σ=0 is a dimension-collapse point and σ=−1.0 a geometric phase transition point; Quantum Chaos Correspondence: The zero-spacing distribution of 491 zeros matches GUE statistics; attains a global minimum at σ=0.5; restoring force F>0 for 100% of zeros; 100% of minimal-action paths lie within σ=0.5±0.1; the matching degree of N(T) reaches 99.8%; ζ-Convexity Theory: For the first 100 non-trivial zeros, V(σ) is strictly decreasing for σ<0.5 and strictly increasing for σ>0.5, and strictly convex within σ=0.5±0.15; interval arithmetic verification is completed via the subinterval strategy and Lipschitz propagation method. This paper honestly reports four critical gaps (G1–G4) and analyzes the chasm between numerical evidence and analytic proof, while demonstrating the original value of this system as an independent number-theoretic framework. Keywords: Riemann Hypothesis, Riemann ζ function, non-trivial zeros, phase geometry, timelike/spacelike structure, quantum chaos, random matrix theory, interval arithmetic, non-associative algebra 摘要 本文提出并系统发展了一套三维数相位几何体系(3DNPG),旨在从几何稳定性角度理解黎曼 ζ 函数非平凡零点的分布规律。该体系的核心思想是:将 ζ 函数在临界带内的行为映射到一个三维相位空间中,通过分析该空间中的类时 / 类空结构、迷向边界、围道积分以及多体耦合等几何对象,揭示临界线 σ=1/2 作为 "最稳定方向" 的深层原因。 通过 A–T 共二十组系统实验,我们建立了以下成果: 代数基石:乘法交换性、莱布尼茨法则精确成立、近似结合性(δ 分量小时误差 O (ε)); 非结合误差控制:证明弱 δ 调制不改变零点拓扑类; 迷向边界软过渡:发现迷向边界是 "软过渡" 而非 "硬奇点",48% 围道点满足 norm<0 但积分不发散; 局部稳定性:σ=0.5 处类时占比为零,σ=1.0 处类空占比 0.798,ρ₁ 附近完全类时; 围道几何与留数理论:积分核为 "近似梯度型",旋度非零源于有限截断欧拉乘积的非解析性; 多体耦合:发现 I_k ∼ k^(−0.89) ≈ 2/k 的普适标度律; 解析延拓:积分函数方程 I (σ) ≈ I (1−σ) 在误差 <3% 下成立,σ=0 为维度坍缩点,σ=−1.0 为几何相变点; 量子混沌对应:491 个零点的间隔分布吻合 GUE 统计,V (σ)=|ζ(σ+iγ)|² 在 σ=0.5 处为全局极小值,100% 零点恢复力 F>0,路径积分最小作用量 100% 落在 σ=0.5±0.1,N (T) 吻合度 99.8%; ζ- 凸性理论:对前 100 个非平凡零点,V (σ) 在 σ<0.5 严格递减、σ>0.5 严格递增,在 σ=0.5±0.15 内严格凸;通过分区间策略和 Lipschitz 传播法完成区间算术验证。 本文同时诚实报告了四个关键缺口(G1–G4),分析了从数值证据到解析证明的鸿沟,并论证了该体系作为独立数论框架的原创价值。 关键词:黎曼假设,黎曼 ζ 函数,非平凡零点,相位几何,类时 / 类空结构,量子混沌,随机矩阵理论,区间算术,非结合代数