Geometric Constraints on the QCD Scale and Strong Coupling from a QTP-Strong Framework

Abstract In this paper, we present a geometric framework designed to determine both the strong coupling strength and the QCD scale from a unified mass–charge structure. We introduce a simple, dimensionless mass–charge identity linking the effective strong charge to the neutral–charged pion mass ratio, ( e / g s ) 2 + ( M π 0 / M π ± ) 2 = 1, which defines a strong-sector mixing angle θ S . Based on a discrete hierarchy of effective charges, the framework predicts a geometric angle of θ S ≈ −14.5° (for g s = 4 e ). This prediction accurately reproduces the observed neutral–charged pion mass ratio and geometrically characterizes the strong interaction as a binding-dominated regime. Building on the hierarchy 1 e →2 e →4 e , the strong fine-structure constant is obtained from a universal geometric relation, α = ( q / q pl ) 2 , in agreement with experimental determinations at the percent level without adjustable parameters. Moreover, the confinement mass scale is derived directly from the proton charge radius as m Λ = ℏ/( r p c ) ≈ m p /4, yielding a value consistent with the phenomenological QCD scale Λ QCD . Taken together, these results indicate that the electromagnetic, weak, and strong interactions can be consistently described within a unified geometric framework at the level of coupling strengths and characteristic mass scales.