Signature Algebras: Low-Dimensional Classification and a Numerical Existence Conjecture at Dimension Four
Lan GuangHeng · Zenodo (CERN European Organization for Nuclear Research) · 2026
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.