What question did this study set out to answer?

To test whether the spacing ratios of low-lying Riemann zeta zeros follow standard Gaussian Unitary Ensemble predictions or converge toward a golden ratio steady-state.

August 16, 2026Open Access

Golden Steady-State as Spectral Attractor: Riemann Zero Spacing Ratios and the Discrete Functional Geometry of Prime Hierarchical Dissipation 黄金稳态作为谱吸引子:黎曼零点间距比与素数层级耗散的离散泛函几何

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Authors

ZLZhongqiang Liu

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Implication

Numerical analysis reveals golden ratio conjugate spacing in low-lying Riemann zeta zeros, suggesting a hierarchical dissipative phase transition distinct from standard random matrix theory.

Key Points

To test whether the spacing ratios of low-lying Riemann zeta zeros follow standard Gaussian Unitary Ensemble predictions or converge toward a golden ratio steady-state.
Analyzed the symmetric spacing ratios for the first 5000 non-trivial zeros of the Riemann zeta function.
Performed statistical hypothesis testing (Z-scores) comparing observed metrics against the Gaussian Unitary Ensemble (GUE) limit of 0.599750 and the golden ratio conjugate (0.618034).
Formulated a discrete functional geometry framework based on scale self-similarity, damped Helmholtz dynamics, and closed hierarchical coupling.
The sample of 5000 non-trivial zeros demonstrated a mean spacing ratio of 0.616335 and a median of 0.623715.
The standard GUE limit hypothesis was rejected at the 5-sigma confidence level (Z = 5.234).
The golden steady-state value remained statistically consistent with the sample (Z = -0.641) and fell inside the 95% confidence interval.

Cite This Study

Zhongqiang Liu (2026) studied this question.

synapsesocial.com/papers/6a817ab4f2fb91fc834aec66 https://doi.org/10.5281/zenodo.21952348

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August 16, 2026Open Access

Golden Steady-State as Spectral Attractor: Riemann Zero Spacing Ratios and the Discrete Functional Geometry of Prime Hierarchical Dissipation 黄金稳态作为谱吸引子:黎曼零点间距比与素数层级耗散的离散泛函几何

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Authors

ZLZhongqiang Liu

Discussion

Member takes

Implication

Numerical analysis reveals golden ratio conjugate spacing in low-lying Riemann zeta zeros, suggesting a hierarchical dissipative phase transition distinct from standard random matrix theory.

Key Points

To test whether the spacing ratios of low-lying Riemann zeta zeros follow standard Gaussian Unitary Ensemble predictions or converge toward a golden ratio steady-state.
Analyzed the symmetric spacing ratios for the first 5000 non-trivial zeros of the Riemann zeta function.
Performed statistical hypothesis testing (Z-scores) comparing observed metrics against the Gaussian Unitary Ensemble (GUE) limit of 0.599750 and the golden ratio conjugate (0.618034).
Formulated a discrete functional geometry framework based on scale self-similarity, damped Helmholtz dynamics, and closed hierarchical coupling.
The sample of 5000 non-trivial zeros demonstrated a mean spacing ratio of 0.616335 and a median of 0.623715.
The standard GUE limit hypothesis was rejected at the 5-sigma confidence level (Z = 5.234).
The golden steady-state value remained statistically consistent with the sample (Z = -0.641) and fell inside the 95% confidence interval.

Cite This Study

Zhongqiang Liu (2026) studied this question.

synapsesocial.com/papers/6a817ab4f2fb91fc834aec66 https://doi.org/10.5281/zenodo.21952348

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