What question did this study set out to answer?

The aim is to establish that the SU(n) hierarchy is generated by the parameter χ = 616 through a unique polynomial.

June 27, 2026Open Access

Dynamic Generation of the SU(n) Hierarchy from χ = 616

Key Points

The aim is to establish that the SU(n) hierarchy is generated by the parameter χ = 616 through a unique polynomial.
Developed a fourth-degree polynomial n(k) with rational coefficients to express the SU(n) hierarchy.
Analyzed connections between polynomial coefficients and Fibonacci numbers.
Investigated the role of χ = 616 as a dynamic parameter for the hierarchy.
All polynomial coefficients are exact rational fractions, confirming the discrete arithmetic structure of the hierarchy.
Noteworthy connections exist between specific coefficients and Fibonacci numbers, demonstrating an underlying mathematical relationship.
Establishment of a direct link between χ and the entire SU(n) hierarchy, showcasing its dynamic nature.

Abstract

This work demonstrates that the entire hierarchy of special unitary groups SU (n) is generated by a single fundamental parameter χ = 616. A unique fourth-degree polynomial is proven to exist: n (k) = (69/8) k⁴ - (777/25) k³ + (63/8) k² + (2029/50) k with rational coefficients that reproduce all key groups of the hierarchy: SU (7) → SU (0) → SU (26) → SU (2) → SU (52) → SU (507) → SU (1905) →. . . Key results: - All polynomial coefficients are exact rational fractions, indicating the discrete arithmetic nature of the hierarchy- Connections are found between coefficients and Fibonacci numbers: 63 = 3 × F₈- Approximate identities connect coefficients to χ = 616 and φ = (1+√5) /2- The canonical chain is established: χ ↔ SU (26) ↔ n (1) = 26 ↔ n (k) ↔ the entire hierarchy- χ acts as a dynamic parameter that fixes the entire infinite hierarchy of gauge groups This work is the second part of the Σ-Ω⁴ Omega Unified Physics trilogy. Contents: - dynamicₕierarchyₐrticleᵣu. pdf — Russian version- dynamicₕierarchyₐrticleₑn. pdf — English version- dynamicₕierarchyₐrticleᵣu. tex — LaTeX source (RU) - dynamicₕierarchyₐrticleₑn. tex — LaTeX source (EN) - DYNAMIC HIERARCHY GENERATION. py — Python implementation- dynamicₕierarchyᵣesults. json — Complete analysis results License: Universal Academic Licence v1. 0 (UAL-v1. 0) Commercial use prohibited without author's written permission. Related work: - SU (0) Operator: Projection onto the Energy-Vacuum Condensate DOI: 10. 5281/zenodo. 20855845

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