What question did this study set out to answer?

To develop an analytic-algebraic framework for complexified three-dimensional algebra and evaluate the spectral properties of the Riemann zeta function within isotropic geometry.

August 19, 2026Open Access

Complexified Three-Dimensional Algebra: From Isotropic Geometry to Golden Polynomial Spectra 复化三维数代数体系:从迷向几何到黄金多项式谱系

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ZLZhongqiang Liu

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Implication

Theoretical analysis reveals isotropic pseudo-zeros linked to the golden ratio within complexified 3D algebra, indicating an independent spectral system for the Riemann zeta function.

Key Points

To develop an analytic-algebraic framework for complexified three-dimensional algebra and evaluate the spectral properties of the Riemann zeta function within isotropic geometry.
Complexified real non-associative algebras and developed midpoint integration with associator corrections on isotropic contours to suppress non-associativity.
Embedded the Riemann zeta function into the complexified three-dimensional algebra and conducted numerical searches for pseudo-zeros across the interval t in [10, 100].
Derived higher-dimensional extensions and large-t asymptotic densities to characterize the resulting golden polynomial spectrum.
Demonstrated that the isotropic norm induces pseudo-zeros precisely governed by the golden ratio inverse phi^-1 = (sqrt(5)-1)/2, uniquely matching n = 3 to Fibonacci structures and infinite dimensions to the critical line coordinate 1/2.
Identified 68 pseudo-zeros in the range t in [10, 100], showing densities positively correlated with standard Riemann zeros and real coordinate distributions biased to the left of the critical line.
Established via asymptotic analysis that pseudo-zero densities remain non-vanishing at large t, forming an independent spectral system.

Cite This Study

Zhongqiang Liu (2026) studied this question.

synapsesocial.com/papers/6a85633c03308d306e2d648b https://doi.org/10.5281/zenodo.21959859

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Implication

Theoretical analysis reveals isotropic pseudo-zeros linked to the golden ratio within complexified 3D algebra, indicating an independent spectral system for the Riemann zeta function.

Complexified Three-Dimensional Algebra: From Isotropic Geometry to Golden Polynomial Spectra 复化三维数代数体系:从迷向几何到黄金多项式谱系

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Cite This Study

Complexified Three-Dimensional Algebra: From Isotropic Geometry to Golden Polynomial Spectra 复化三维数代数体系:从迷向几何到黄金多项式谱系

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Implication

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