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Main Author:	Ajaib, M. Adeel
Format:	Preprint
Published:	2025
Subjects:	High Energy Physics - Phenomenology
Online Access:	https://arxiv.org/abs/2512.02059
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author	Ajaib, M. Adeel
author_facet	Ajaib, M. Adeel
contents	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.
format	Preprint
id	arxiv_https___arxiv_org_abs_2512_02059
institution	arXiv
publishDate	2025
record_format	arxiv
spellingShingle	Emergent Grand Unified Structure in a 4 x 4 Nilpotent Matrix Algebra Ajaib, M. Adeel High Energy Physics - Phenomenology 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.
title	Emergent Grand Unified Structure in a 4 x 4 Nilpotent Matrix Algebra
topic	High Energy Physics - Phenomenology
url	https://arxiv.org/abs/2512.02059

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