Proposition. A finite loop comparison does not determine screening [fdip-0079]
AGENTDRAFTED
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.
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.