What question did this study set out to answer?

Construct a self-consistent axiomatic three-dimensional number algebra that unifies the complex plane with vertical hyperbolic phase space for applications in geometry and number theory.

August 16, 2026Open Access

Three-Dimensional Number Algebra System — A Non-Associative Algebra Unifying Complex Plane and Hyperbolic Phase Space with Applications in Differential Geometry and Number Theory

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Authors

ZLZhongqiang Liu

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Implication

Theoretical mathematical framework establishes a non-associative three-dimensional number system, indicating new analytical tools for differential geometry and the Riemann hypothesis.

Key Points

Construct a self-consistent axiomatic three-dimensional number algebra that unifies the complex plane with vertical hyperbolic phase space for applications in geometry and number theory.
Formulated ordered triples into an axiomatic algebraic framework with closed formulas for arithmetic operations, conjugates, algebraic norms, and Euclidean norms.
Classified octants, subalgebras, and zero-divisor singular regions while evaluating compatibility with Riemannian metric tensors and Laplace–Beltrami operators on 3D manifolds.
Demonstrated that the 3D algebra fully inherits standard complex number operational properties while providing independent vertical phase freedom.
Applied the algebraic system to construct the full 3D phase manifold of the Riemann zeta function and model critical line convexity potential wells.

Cite This Study

Zhongqiang Liu (2026) studied this question.

synapsesocial.com/papers/6a817a33f2fb91fc834adef0 https://doi.org/10.5281/zenodo.21935515

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August 16, 2026Open Access

Three-Dimensional Number Algebra System — A Non-Associative Algebra Unifying Complex Plane and Hyperbolic Phase Space with Applications in Differential Geometry and Number Theory

View Full Paper

Ask AI

Bookmark

Authors

ZLZhongqiang Liu

Discussion

Member takes

Implication

Theoretical mathematical framework establishes a non-associative three-dimensional number system, indicating new analytical tools for differential geometry and the Riemann hypothesis.

Key Points

Construct a self-consistent axiomatic three-dimensional number algebra that unifies the complex plane with vertical hyperbolic phase space for applications in geometry and number theory.
Formulated ordered triples into an axiomatic algebraic framework with closed formulas for arithmetic operations, conjugates, algebraic norms, and Euclidean norms.
Classified octants, subalgebras, and zero-divisor singular regions while evaluating compatibility with Riemannian metric tensors and Laplace–Beltrami operators on 3D manifolds.
Demonstrated that the 3D algebra fully inherits standard complex number operational properties while providing independent vertical phase freedom.
Applied the algebraic system to construct the full 3D phase manifold of the Riemann zeta function and model critical line convexity potential wells.

Cite This Study

Zhongqiang Liu (2026) studied this question.

synapsesocial.com/papers/6a817a33f2fb91fc834adef0 https://doi.org/10.5281/zenodo.21935515

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