Relation between α(E) (y-axis) for the peak flux and particle energy E(x-axis) with the evolution of the theoretical spectral and propagation parameters, γ, κ₀, and β. Panels (a)–(c) explore the variations when β equals 1, while panels (d)–(f) present the results for γ at 3.75. In each panel, the pink diamonds represent the α within four energy ranges, derived from the mean value of Group B, and the error bars are the derived standard deviation.