Paper CLI — Measure-rigorous F₂-CFN ↔ SU(N) Wilson dictionary

Title: Paper CLI — Measure-rigorous F₂-CFN ↔ SU(N) Wilson dictionary

Author: Alexander Novickis (alex.novickis@gmail.com)

The companion paper CIII v3 establishes a positive mass gap for the branched 6-point correlator on the F₂-CFN-decomposed SU(2) Wilson measure. The Cho-Faddeev-Niemi (CFN) decomposition splits the SU(N) link variable into a colour-direction field $\hat n$ valued in the $F_2$ flag manifold, a transverse field $X_\mu$, and a residual U(1) phase $\rho$. While the decomposition is well-established at the physics level (Cho 1980; Faddeev-Niemi 1999; Shabanov 1999), its measure-theoretic equivalence to the standard Wilson lattice measure has never been established at the constructive-QFT level. This paper closes the bridge. We construct a measurable Jacobian $J = \Delta_{\rm FP} \cdot \mathcal V_{\rm Pl}$ (Faddeev-Popov determinant times Plücker volume form on $F_2$) and prove that the change of variables $d\mu_W^{\rm gf} = J\, d\mu_{F_2\text{-CFN}}$ is an equivalence of finite-lattice measures. The continuum-limit Osterwalder-Schrader axioms (OS0 temperedness, OS1 hyper-cubic covariance, OS3 reflection positivity) are preserved under the change of variables, reducing to a single uniform-in-$(L,a)$ Jacobian bound (Eq. 5.1) which we close in two steps: (i) $\mathcal V_{\rm Pl}$ is a smooth positive density on the compact flag manifold $F_2$, with explicit numerical extrema $V_{\min}^{SU(3)} \approx 0.5$, $V_{\max}^{SU(3)} \approx 2.0$; (ii) $\Delta_{\rm FP}$ factorises as $\Delta_0(\beta) \cdot \Delta_1$ with the configuration-independent $\Delta_0$ cancelling in normalised Schwinger functions, and $\Delta_1 \to 1$ as $a \to 0$ along the renormalised trajectory with explicit density bound $(1/L^4)|\log \Delta_1| \leq 0.78$ at $\beta=185$. OS2/OS4/OS5 are inherited via Paper CL. As a corollary, the mass-gap result of CIII v3 transfers to the standard gauge-fixed Wilson SU(N) measure, completing the programme-internal bridge from the CFN-decomposed measure to the literature-standard object.

Series: Paper CLI in the Hopf Soliton Programme