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G. ’t Hooft,Topology of the Gauge Condition and New Confinement Phases in Non-Abelian Gauge Theories, Nucl. Phys. B190, 455 (1981). DOI: 10.1016/0550- 3213(81)90442-9 Supplemental Material for: Systematic Extraction of Exact Yang-Mills Solutions via Algebraic Tensor Ring Decomposition Yu-Xuan Zhang 1 and Jing-Ling Chen 2,∗ 1School of Physics, Nankai University, Tianjin 300071, People’s Republic of China 2Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China (Dated: May 8, 2026) CONTENTS I. Derivations for RelativisticSU(2) Color Waves 1 II. Derivations for Dynamical Helical Flux Tubes 4 III. Derivations forSU(3) Dynamics and Kinetic Cancellation 7 IV. Derivations for Semiclassical Quantization 9 A. SU(2) Background Mass Matrix Diagonalization 9 B. SU(3) Background Mass Matrix and Positive Definiteness 10 V. The Spin-Magnetic Coupling and Evasion of Savvidy Instability 11 A. SU(2) Relativistic Color Waves 11 B. SU(3) Chaotic Resonances 11 C. Asymptotic Eigenvalues of the Dynamical Flux Tube 12 This Supplemental Material provides the exhaustive, step-by-step mathematical derivations supporting the claims made in the main text. It details the explicit calculation of the non-Abelian field strength tensors in locally rotated frames, the algebraic projection of the Yang-Mills equations into differential ideals, the exact quotient ring evaluations, and the semiclassical stability analysis. I. DERIVATIONS FOR RELAT
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