The Multiscale Geometry of Riemann Zeta Zeros: Pairwise Scale Structure and Low-Dimensional Organization

Abstract

The nontrivial zeros of the Riemann zeta function have been extensively studied through the statistical properties of their normalized spacings. In this work, we investigate a different structural representation based on the multiscale behavior of consecutive Riemann zeta-zero spacings. Starting from the sequence of consecutive zero spacings, we construct a locally normalized representation and evaluate a recovery observable over a range of finite window scales. Pairwise comparisons between these scales define a geometric map whose coordinates are given by the mean scale and the scale asymmetry. We first identify a compact CORE representation that captures most of the observed pairwise variation. We then examine nonlinear scale–asymmetry interactions and find that a low-order interaction provides a consistent predictive extension of the CORE model across repeated out-of-sample tests. A complementary principal-component analysis shows that the nonlinear interaction sector is strongly low-dimensional, with its leading component accounting for approximately 98.25% of its variance. However, target-permutation controls indicate that the observed improvements in raw R² and mean-squared error are not, by themselves, specific to the original target structure under the tested conditional null model. The results therefore provide evidence for a low-dimensional organization within the proposed multiscale representation, rather than a universal law for Riemann zeta zeros or a direct consequence for the Riemann Hypothesis.