What does this research mean for the field?

Signature algebras are a newly introduced class of algebraic structures lacking universal left-distributivity and commutativity, which possess complete classifications for dimensions two and three, lack valid representations on flat algebras for N ≥ 5, and show strong numerical evidence for existence at N = 4. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.ESTABLISHES_NEW_DIRECTION.

What question did this study set out to answer?

This research aims to establish a classification of signature algebras and investigate the existence of valid representations in various dimensions.

July 22, 2026Open Access

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Signature Algebras: Low-Dimensional Classification and a Numerical Existence Conjecture at Dimension Four

LGLan GuangHeng

Key Points

This research aims to establish a classification of signature algebras and investigate the existence of valid representations in various dimensions.
Introduced signature algebras with specific algebraic properties
Classified two- and three-dimensional nilpotent representations
Conducted numerical experiments to support existence conjecture for N=4.
Completed classification of two- and three-dimensional nilpotent representations.
Showed that no valid representations exist on flat algebras for N ≥ 5.
Provided numerical evidence of loss ~ 3.4e-12 suggesting a valid representation exists at dimension N=4.

Abstract

We introduce signature algebras, a new class of algebraic structures where the "addition" is commutative and associative, the "multiplication" is associative and right-distributive, while neither commutativity nor left-distributivity holds universally. Main results:- Complete classification of two- and three-dimensional nilpotent representations- Proof that no valid representations exist on flat algebras for N ≥ 5- Numerical evidence (loss ~ 3.4e-12) supporting the conjecture that a non-projective valid representation exists at N = 4- Graded generalization with algebraic proof of the Pauli exclusion principle- Formal derivation of the continuum limit under translational invariance All symbolic computations are verified with SymPy and numerical experiments are fully reproducible via the supplementary Python scripts.

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Signature Algebras: Low-Dimensional Classification and a Numerical Existence Conjecture at Dimension Four

Key Points

Abstract

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