AI 中文总结
本研究开发移动轮廓汉克尔框架编码黎曼Ξ函数局部零点构型，推导相关准则与惯性公式，验证了轮廓求积等方法的有效性，为黎曼假设提供局部汉克尔正性表述。
AI 中文摘要
我们开发了一种移动轮廓汉克尔框架，用于编码黎曼Ξ函数的局部零点构型。与轮廓相关的全纯坐标下，对数导数Ξ'/Ξ的加权轮廓积分被识别为支撑在被包围零点的坐标像上的有限原子测度的幂矩。该表示给出了精确的无零点和秩准则，且对于与共轭兼容的轮廓-坐标对，存在惯性公式：一旦矩阵阶数至少等于不同坐标节点的数量，负指数就等于不同非实共轭对的数量。因此，黎曼假设可表述为局部有限维汉克尔正性，不过独立于零点集确立该正性仍未解决。当轮廓移动时，汉克尔矩阵在零交叉之间通过连续同余流演化，并在交叉事件处经历有限秩跳跃：孤立零点产生带符号的秩1跳跃，而非实共轭对在与该对相遇的实轴圆形扫描中产生秩2不定事件。消除连续坐标漂移后，可得到分段常数矩阵过程，从中可恢复交叉坐标、重数和零点位置。数值实验验证了轮廓求积、指示场、连续流、交叉特征及恢复程序。
英文摘要
We develop a moving-contour Hankel framework for encoding local zero configurations of the Riemann $Ξ$-function. Weighted contour integrals of the logarithmic derivative $Ξ'/Ξ$, expressed in a holomorphic coordinate associated with the contour, are identified with the power moments of a finite atomic measure supported at the coordinate images of the enclosed zeros. This representation yields exact zero-free and rank criteria and, for conjugation-compatible contour-coordinate pairs, an inertia formula: once the matrix order is at least the number of distinct coordinate nodes, the negative index equals the number of distinct nonreal conjugate pairs. Consequently, the Riemann hypothesis admits a local finite-dimensional Hankel-positivity formulation, although establishing this positivity independently of the zero set remains unresolved. As the contour moves, the Hankel matrix evolves by a continuous congruence flow between zero crossings and undergoes finite-rank jumps at crossing events. An isolated zero produces a signed rank-one jump, whereas a nonreal conjugate pair produces a rank-two indefinite event in a real-axis circular scan that meets the pair. Removing the continuous coordinate drift yields a piecewise-constant matrix process from which crossing coordinates, multiplicities, and zero locations can be recovered. Numerical experiments validate the contour quadrature, indicator fields, continuous flow, crossing signatures, and recovery procedure.
