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arXiv · 2512.02059

Emergent Grand Unified Structure in a 4 x 4 Nilpotent Matrix Algebra

M. Adeel Ajaib
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Abstract

We show that nilpotent matrices that yield the Schrodinger equation from its first order form encode the fingerprints of grand unified theories. We perform a rigorous search for all such nilpotent matrices and find that the resulting matrices naturally organize into suggestive group theoretic structures without any other a priori assumptions. The antisymmetric sector consists of three groups of sixteen matrices, each of which further splits as 16 = 12 + 4 and exhibits unique characteristics in the step potential scattering problem. The symmetric zero-diagonal sector also forms three families, mirroring the quark-lepton decomposition of the Pati-Salam model. These results may help answer why there are three families of fermions and also demonstrate that the 4 x 4 matrix algebra is a compact, nontrivial shadow of the SO(10) embedding, with fermion-like and gauge-like subspaces.

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M. Adeel Ajaib. 2025-11-28. Emergent Grand Unified Structure in a 4 x 4 Nilpotent Matrix Algebra. https://arxiv.org/abs/2512.02059

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