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August 19, 2026Open Access

Orthogonal-Zero Non-Associative Algebra (𝕋-Algebra) β€” Theory IV of the 3D Number System

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Authors

ZLZhongqiang Liu

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Implication

Theoretical analysis proves that three dimensions represent the minimal real algebra satisfying the orthogonal-zero condition, establishing foundational rules for non-associative 3D number systems.

Key Points

  • Construct an axiomatic system for 3D number algebra (𝕋-algebra) and identify the minimal-dimensional real algebra satisfying the orthogonal-zero condition.
  • Axiomatically developed the theoretical framework of 𝕋-algebra, deriving inverse formulas, mid-root definitions, module decompositions, and high-dimensional extensions.
  • Verified analytical algebraic proofs and theoretical properties through numerical experiments implemented in Python.
  • Proved via the Classification Theorem that the minimal-dimensional real algebra satisfying the orthogonal-zero condition is exactly three-dimensional.
  • Successfully established mathematical formulations for inverse operations, left/right module decomposition, and high-dimensional algebra extensions.

Cite This Study

Zhongqiang Liu (2026) studied this question.

synapsesocial.com/papers/6a85636403308d306e2d67adhttps://doi.org/10.5281/zenodo.21966391
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