Proposition. A finite loop comparison does not determine screening [fdip-0079]
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  • November 1, 2026

On a square torus of periods at least three, use the joint law with \(G=\{1,-1\}\) acting by multiplication on the real field \(\phi \in \mathbb R^V\), unit weights, \(m>0\) and \(\beta =0\). For any simple nonempty closed edge cycle \(C\), its Wilson product \(W_C(U)=\prod _{e\in C}U_e\) has \[\langle W_C\rangle _{\mathrm {matter}}>0,\qquad \langle W_C\rangle _{\mathrm {pure}}=0.\] In particular, the strict positivity in the three-by-three example has a direct proof independent of rounding the determinant sums.

Proof. 

  • November 1, 2026

Put \(D=4+m^2\). For the real scalar convention, the joint Boltzmann factor at zero bare gauge coupling is \[\exp \left (-\frac D2\sum _v\phi _v^2\right ) \prod _{e\in E}\exp (U_e\phi _{s(e)}\phi _{h(e)}).\] Average the independent edge signs before integrating the matter. Since \[\frac 12\sum _{u=\pm 1}e^{ut}=\cosh t,\qquad \frac 12\sum _{u=\pm 1}u e^{ut}=\sinh t,\] the Wilson expectation is \(I_C/I_{\varnothing }\), where \[\begin {aligned} I_C={}&\int _{\mathbb R^V}e^{-\frac D2\sum _v\phi _v^2} \prod _{e\in C}\sinh (\phi _{s(e)}\phi _{h(e)})\\ &\qquad \qquad \cdot \prod _{e\notin C} \cosh (\phi _{s(e)}\phi _{h(e)})\,d\phi . \end {aligned}\] For \(C=\varnothing \), every edge contributes a hyperbolic cosine. For nonzero vertex coordinates, the sign of the numerator's product of hyperbolic sines is \[\prod _{e\in C}\operatorname {sgn}(\phi _{s(e)}\phi _{h(e)}) =\prod _{v\in C}\operatorname {sgn}(\phi _v)^2=1.\] Each cycle vertex occurs twice. All other factors are strictly positive, so the numerator integrand is positive off coordinate hyperplanes. Its absolute value is bounded by the denominator integrand because \(|\sinh t|\le \cosh t\). Positive mass makes the latter integrable by the joint-law bound. Both integrals are therefore finite and strictly positive. In the pure law, independent uniform edge signs give zero expectation for any nonempty product of distinct edges, proving the comparison.

Integrating matter has changed the connection law, so the exact pure-gauge formula cannot be reused unchanged. On tori with periods at least 6, the closed-walk calculation gives a separate controlled large-mass explanation: matter generates a positive plaquette coefficient and higher loop terms. Its order \(D^{-6}\) remainder is not being applied to the three-by-three example, where short winding cycles occur.

Neither the finite comparison nor the positive plaquette term establishes an asymptotic area law, a perimeter law or string breaking. Such a claim needs a specified family of volumes and loops, a limiting regime and control of the higher terms. Gliozzi and Rago study a different system: a three-dimensional two-element gauge group coupled to Ising matter taking only the values \(\pm 1\), rather than an unbounded Gaussian field. Their Section 3, equation (12), fixes that action; their analysis of Wilson-loop overlap with a broken-string state is model-specific context [gliozzi2004overlap, pp. 2, 5]. It does not supply a screening theorem for the present Gaussian law.