The Weak-Field Ultraviolet Limit of Fried's QCD Functional Formalism

We examine the weak-field ultraviolet limit of Fried's QCD functional
formalism. Working in the convention
\[
S_{\rm YM}[A]
=
\frac{1}{4g^2}
\int d^4x\,
\mathcal F_{\mu\nu}^a[A]\mathcal F_{\mu\nu}^a[A],
\]
we use the background-field method to show that the Halpern-field
representation reconstructs the standard Yang--Mills quadratic fluctuation
operator in the short-distance limit. The Halpern saddle is
\[
\chi_{\mu\nu}^a
=
\frac{i}{g^2}\mathcal F_{\mu\nu}^a[B],
\]
and fluctuations around this saddle reconstruct the usual quadratic kinetic
term for the quantum gauge field. Background-field gauge fixing supplies the
standard Faddeev--Popov ghost determinant, while the explicit quark-loop
determinant gives the usual fermionic screening contribution. The resulting
one-loop effective action reproduces the universal QCD beta-function
coefficient
\[
b_0
=
\frac{11}{3}N_c-\frac{2}{3}N_f .
\]
Thus Fried's nonperturbative functional rearrangement preserves the
asymptotically free ultraviolet limit of QCD. This result should be understood
as a UV consistency theorem for the formalism, not as a proof of confinement,
Pomeron dynamics, or the full infrared behavior of QCD.