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

M. Adeel Ajaib

TL;DR

This work proposes a discrete, 4×4 nilpotent matrix framework for nonrelativistic and relativistic quantum dynamics that encodes the fingerprints of grand unified theories. By enforcing nilpotency ($\eta^2=0$), transpose symmetry, and a discrete alphabet, the authors construct 48 antisymmetric and 48 symmetric zero-diagonal matrices that organize into three 16-plets with a universal (12+4) Pati-Salam structure, reproducing an $\mathfrak{su}(2)_L \oplus \mathfrak{su}(2)_R$ chiral sector and revealing a representation-level correspondence to SO(10). The framework yields three generation-like sectors with distinct dispersion and energy-dependent spin-transport signatures, and a full 432-matrix symmetric sector that extends the fermion-like content with additional Pauli-pattern families. Experimental signatures in spin-polarized electron scattering are proposed as concrete tests of the predicted generation structure and the emergent SO(10)-like correspondence, offering a finite algebraic route to understanding fermion multiplicity without invoking continuous extra symmetries.

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.

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

TL;DR

This work proposes a discrete, 4×4 nilpotent matrix framework for nonrelativistic and relativistic quantum dynamics that encodes the fingerprints of grand unified theories. By enforcing nilpotency (), transpose symmetry, and a discrete alphabet, the authors construct 48 antisymmetric and 48 symmetric zero-diagonal matrices that organize into three 16-plets with a universal (12+4) Pati-Salam structure, reproducing an chiral sector and revealing a representation-level correspondence to SO(10). The framework yields three generation-like sectors with distinct dispersion and energy-dependent spin-transport signatures, and a full 432-matrix symmetric sector that extends the fermion-like content with additional Pauli-pattern families. Experimental signatures in spin-polarized electron scattering are proposed as concrete tests of the predicted generation structure and the emergent SO(10)-like correspondence, offering a finite algebraic route to understanding fermion multiplicity without invoking continuous extra symmetries.

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.

Paper Structure

This paper contains 15 sections, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Matrix Construction and Multiplicity Structure
  3. Quantum Dynamics and Scattering
  4. Dispersion Relations
  5. Transmission Patterns and Generation Structure
  6. SO(10) Correspondence
  7. Pati-Salam Decomposition: (12+4) Structure
  8. Emergent Chiral Structure
  9. Lie Algebra Structure and Nilpotent Constraints
  10. Representation-Level Correspondence
  11. Higher Symmetric Nilpotent Sectors: Full 432-Matrix Structure
  12. Experimental Signatures
  13. Conclusion
  14. Acknowledgments
  15. Transmission Coefficients

Figures (1)

  • Figure 1: Relativistic transmission coefficients for spin-up electrons incident on a step potential for the three antisymmetric matrix families as functions of energy $E/(V_0+m)$. $T_1$ (red): transmitted as spin-up (no flip). $T_2$ (blue): transmitted as spin-down (spin-flip). $T_{\text{int}}$ (green): interference term. (a) Generation 1 (zero diagonal blocks) shows sharp threshold with strong spin-flip activity near threshold, characteristic of light particles. (b) Generation 2 (real diagonal entries) displays intermediate behavior with suppressed spin-flip. (c) Generation 3 (complex diagonal entries) exhibits smooth threshold with most suppressed spin-flip, characteristic of heavy particles.