A finite SU(2) Wilson lattice and its gauge-invariant transfer sector

When every link is $I_2$, every plaquette product is $I_2$. Each action term is $\beta (1-\frac{1}{2}{\mathrm{ReTr}}_2{I}_2)=0$, so $S_{\mathrm{\Lambda }}=0$; the ordered product around $C$ is $I_2$, hence $W_C=1$. This evaluates the observable on one configuration only, not its probability or expectation.

The coordinate projection $U\mapsto u_0(U)$ is continuous for the finite-product topology, and matrix multiplication, inversion, and trace are continuous; hence $W_C:{\mathcal{C}}_{\mathrm{\Lambda }}\to \mathbb{R}$ is continuous and Borel-measurable for the stated product Borel sigma-algebra. For a closed path $C=(x_0,{\mathrm{\ell }}_1,x_1,\dots ,{\mathrm{\ell }}_m,x_m)$ with $x_m=x_0$, each oriented factor transforms by the gauge matrix at its start and the inverse gauge matrix at its end. Adjacent factors cancel, leaving $U_C\mapsto {\mathrm{\Omega }}_{x_0}{U}_C{\mathrm{\Omega }}_{x_0}^{-1}$; cyclicity of trace proves gauge invariance. By [F3], for $V\in SU(2)$ one has $\frac{1}{2}{\mathrm{ReTr}}_{2}V=\mathrm{Re}\alpha \in [-1,1]$. Thus $W_C$ is bounded by one in absolute value.

The choices $N_s=N_t=3$, $a>0$, and $g_0>0$ satisfy the finite-lattice hypotheses of [F1]. Applying [F1] gives a finite positive partition function, a defined normalized expectation for this bounded $W_C$, a bounded transfer operator preserving the Gauss sector, and a nonzero constant vector in that sector.

The same application of [F1] gives that the cyclic product of the one-step kernels integrates to the full periodic Euclidean Wilson weight, and that inserting $W_C$ on the chosen slice yields its stated finite-volume expectation. This is a fixed-lattice calculation; it contains no continuum or Lorentzian spectral assertion.