Finite-Regulator Confinement Scales in Fried's QCD Functional Formalism

We study finite confinement-scale candidates within Fried’s nonperturbative QCD functional for-malism. The starting point is the local color-Lorentz matrix structure produced by the exact reor-ganization of gluonic functional integrations, with effective locality entering as a central structuralresult of the formalism. After introducing a finite Schur/eigenvalue regulator, we define a transversematrix profileΦΛ(b) =ZMΛdΩΛ exp[−V (Λ)+ (ΩΛ; ξ)]K(b; ΩΛ).Under compactness, stability, uniform finite-moment, nontriviality, and non-collapse hypotheses, weprove that this profile has a finite and nonzero second moment,0 < R2Λ < ∞.The inverse scale MΛ = R−1Λ defines a finite-regulator confinement-mass candidate. A correspondingstring-tension candidate may be written dimensionally as σΛ ∼ M 2Λ, or structurally as g2CF IΛ[ΦΛ].The result is not a proof of continuum confinement or of a Wilson-loop area law. Rather, it es-tablishes a finite-regulator scale theorem that supports the Fried–Tsang confinement-scale proposaland provides an intermediate step toward a Wilson-loop confinement analysis.