逆谱几何中的隐藏戴森普适性
Hidden Dyson Universality in Inverse-Spectral Geometry
AI总结：
该研究在逆谱几何中揭示了戴森普适性的隐藏表现，通过 dressing 变换结合距离壳矩诊断，证实黎曼 ζ 函数非平凡零点的重构算子具有 GUE 特性。
AI中文摘要：
戴森普适性通常表现为局部本征值统计特性。本文证明其特征在非线性逆谱重构中依然存在，并在重构算子的矩阵几何中重现。利用 dressing 变换，我们将每个展开谱映射为固定简谐振子的形变 $f(x)$，并在公共振子基下表示为 $F_{mn}=\bra m|f|n\ket$。我们将矩阵元权重分解为固定距离 $d=|m-n|$ 的壳层，对应参考振子的能量转移通道，并通过距离壳矩表征所得分布。经高斯 β 系综独立校准，这些矩随 β 平滑变化，可区分 GOE、GUE 和 GSE。固定该校准后，将相同诊断应用于黎曼 ζ 函数的非平凡零点，结果表明重构算子属于 GUE 范畴。因此，零点的 GUE 特性并非通过输入能级的直接统计恢复，而是通过重构算子的距离分辨几何实现。
英文摘要：
Dyson universality typically manifests itself in local eigenvalue statistics. Here we show that its signature survives a nonlinear inverse-spectral reconstruction and reappears in the matrix geometry of the reconstructed operator. Using a dressing transformation, we map each unfolded spectrum to a deformation $f(x)$ of a fixed harmonic oscillator and represent it in the common oscillator basis by $F_{mn}=\bra m|f|n\ket$. We resolve the matrix-element weight into shells of fixed distance $d=|m-n|$, corresponding to the energy-transfer channels of the reference oscillator, and characterize the resulting distribution by distance-shell moments. Independently calibrated on Gaussian $β$-ensembles, these moments vary smoothly with $β$ and distinguish the GOE, GUE, and GSE. With this calibration fixed, applying the same diagnostic to the nontrivial zeros of the Riemann zeta function places the reconstructed operators in the GUE sector. Thus, the GUE character of the zeros is recovered not through direct statistics of the input levels, but from the distance-resolved geometry of the reconstructed operator.
