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Construct and analyze a unified family of commutative, non-associative algebras from 1D to 9D to systematically characterize dimensional scaling laws.

August 19, 2026Open Access

Dimensional Scaling Study: Non-Associative Algebras 1D–9D 非结合代数 1D–9D 维度标度系统性研究报告

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ZLZhongqiang Liu

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Implication

Theoretical analysis demonstrates algebraic signature scaling in commutative non-associative algebras from 1D to 9D, revealing dimensional transitions and non-associative behavior.

Key Points

Construct and analyze a unified family of commutative, non-associative algebras from 1D to 9D to systematically characterize dimensional scaling laws.
Constructed unified algebraic signatures (p,q) where p = ceil((d+1)/2) positive and q = floor((d-1)/2) negative basis elements across dimensions 1D through 9D.
Applied standardized cross-dimensional multiplication rules defining interactions among the identity element, positive basis elements, and negative basis elements.
Confirmed that 1D and 2D systems remain associative, whereas non-associativity first emerges in 3D with signature (2,1).
Demonstrated systematic expansion of negative basis elements up to dimension 9 with signature (4,5), establishing structural baselines for non-associative strength and isotropic scaling.

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Zhongqiang Liu (2026) studied this question.

synapsesocial.com/papers/6a858aaf03308d306e2d7f1d https://doi.org/10.5281/zenodo.21979658

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Implication

Theoretical analysis demonstrates algebraic signature scaling in commutative non-associative algebras from 1D to 9D, revealing dimensional transitions and non-associative behavior.

Dimensional Scaling Study: Non-Associative Algebras 1D–9D 非结合代数 1D–9D 维度标度系统性研究报告

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Dimensional Scaling Study: Non-Associative Algebras 1D–9D 非结合代数 1D–9D 维度标度系统性研究报告

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