A strict polymer majorant gives a zero-free finite partition function

Write $f_{ij}=-{\mathbf{1}}_{\{{\gamma }_i\sim {\gamma }_j\}}$. Reflexivity makes ${\prod }_{i<j}(1+f_{ij})$ zero when a tuple repeats a polymer, and otherwise this product is one exactly for a compatible subset. Hence $Z_V(t)={\sum }_{n\ge 0}\frac{t^n}{n!}{\sum }_{\vec{\gamma }\in V^n}({\prod }_i{w}_{{\gamma }_i}){\prod }_{i<j}(1+f_{ij})$ as a formal series. Expanding the pair product gives all simple graphs. A set partition of $[n]$ is a finite family of nonempty pairwise-disjoint blocks whose union is $[n]$; write ${\mathrm{\Pi }}_n$ for the set of such partitions. This set is finite since it is contained in $\mathcal{P}(\mathcal{P}([n]))$. The connected components of each finite graph give its unique partition into connected graph blocks. Each graph's edge weight factors over these blocks. The coefficient of $t^n$ in $\exp (L_V(t))$ is $\frac{1}{n!}{\sum }_{\pi \in {\mathrm{\Pi }}_n}{\sum }_{\vec{\gamma }\in V^n}({\prod }_{B\in \pi }{U}_{\mathrm{\mid }B\mathrm{\mid }}({\vec{\gamma }}_B))({\prod }_{i=1}^n{w}_{{\gamma }_i})$: for ordered block sizes $m_1,\dots ,m_k$, assigning the $n$ labels to those blocks contributes $n!\mathrm{/}(m_1!\cdots m_k!)$, and the exponential's factor $1\mathrm{/}k!$ removes the ordering of the $k$ blocks. These factors leave exactly the connected-component expansion of the all-graph coefficient. Thus $Z_V$ and $\exp (L_V)$ have identical formal coefficients.

Let $d_n$ be the coefficient of $t^n$ in $L_V$, and apply [F2] to the finite polymer set $V$ with activities $r{w}_{\eta }$. Since $r>0$, [F5] gives $\mathrm{\mid }r{w}_{\eta }\mathrm{\mid }=r\mathrm{\mid }{w}_{\eta }\mathrm{\mid }$, so [A1] implies the rooted criterion. For each fixed root ${\gamma }_1$, the $n$th nonnegative term of that rooted sum is at most $e^{a_{{\gamma }_1}}$. The finite-sum triangle inequality, multiplicativity in [F5], and agreement of the complex modulus with real absolute value give $r^n\mathrm{\mid }{d}_n\mathrm{\mid }\le \frac{1}{n!}{\sum }_{{\gamma }_1\in V}r\mathrm{\mid }{w}_{{\gamma }_1}\mathrm{\mid }{\sum }_{{\gamma }_2,\dots ,{\gamma }_n\in V}\mathrm{\mid }{U}_n({\gamma }_1,\dots ,{\gamma }_n)\mathrm{\mid }{\prod }_{j=2}^n\mathrm{\mid }r{w}_{{\gamma }_j}\mathrm{\mid }\le \frac{1}{n}{\sum }_{\gamma \in V}r\mathrm{\mid }{w}_{\gamma }\mathrm{\mid }{e}^{a_{\gamma }}$. Thus on $\mathrm{\mid }t\mathrm{\mid }\le s<r$, the series is dominated by a constant times ${\sum }_{n\ge 1}(s\mathrm{/}r{)}^n\mathrm{/}n$, and its derivative series by a constant times ${\sum }_{n\ge 1}(s\mathrm{/}r{)}^{n-1}$. Both converge uniformly on every such closed subdisc.

By [F4], $L_V$ differentiates term by term throughout $\mathrm{\mid }t\mathrm{\mid }<r$. For each fixed $t$ with $\mathrm{\mid }t\mathrm{\mid }<r$, step 1.2 gives ${\sum }_{n\ge 1}\mathrm{\mid }{d}_n\mathrm{\mid }\mathrm{\mid }t{\mathrm{\mid }}^n<\mathrm{\infty }$, so the total absolute sum after expanding $\exp (L_V(t))$ is at most ${\sum }_{k\ge 0}({\sum }_{n\ge 1}\mathrm{\mid }{d}_n\mathrm{\mid }\mathrm{\mid }t{\mathrm{\mid }}^n{)}^k\mathrm{/}k!<\mathrm{\infty }$. The products may therefore be regrouped by total degree; their coefficients are the formal coefficients identified in step 1.1, so $Z_V(t)=\exp (L_V(t))$ on the disc. By [F3], $\exp (L_V(t))\exp (-L_V(t))=1$, so $Z_V(t)$ is nonzero and $L_V$ is its analytic logarithm with $L_V(0)=0$.