We analyze the moduli space integrand (39) of the Solomon functional. Applying the uniform energy bound and the wedge region bound, the first term is bounded by

$$|\text{Im}\int_{\Sigma}e^{-i\hat{\theta}}F\omega|\leq\int_{\Sigma}|F|\omega\leq\frac{\text{Re}(F(p))}{\sin\epsilon}\int_{\Sigma}\omega\leq\frac{\text{Re}(F(p))}{\sin\epsilon}A(N+1).$$

More intrinsically $F(p)$ defines the complex valued volume form $\tilde{\Omega}_{L_{0}}$ on the moduli space, hence

$$|\text{Im}\int_{\Sigma}e^{-i\hat{\theta}}F\omega|\leq\frac{1}{\sin\epsilon}A(N+1)\text{Re}(\tilde{\Omega}_{L_{0}}).$$

(46)

The other two terms in (39) are rewritten as a sum of contributions from intersection points in (38). As in Prop. 3.41, we consider holomorphic polygons whose boundary $\partial\Sigma$ encounters in the clockwise order $p_{N-1}\in CF^{1}(L_{N},L_{N-1}),\ldots$, $p_{1}\in CF^{1}(L_{2},L_{1})$, $p=p_{0}\in CF^{0}(L_{1},L_{0})$, $q=p_{N}\in CF^{0}(L_{0},L_{N})$ juxaposed possibly by more degree one self intersections $b_{i}$ of $L_{i}$. (The other cases, where $\partial\Sigma$ misses some Lagrangians, can be handled completely similarly.) We first deal with these extra self intersections. Using the wedge region bound, and the Novikov positivity requirement,

$$|\text{Im}\sum_{i}\sum_{b_{i}}e^{-i\hat{\theta}}Ff_{L_{i}}|^{+}_{-}(b_{i})|\leq\sum_{i}\sum_{b_{i}}|F(b_{i})|f_{L_{i}}|^{+}_{-}(b_{i})\leq\frac{\text{Re}(F(p))}{\sin\epsilon}\sum_{i}\sum_{b_{i}}f_{L_{i}}|^{+}_{-}(b_{i}).$$

By Lemma 3.41, we have

$$|\text{Im}\sum_{i}\sum_{b_{i}}e^{-i\hat{\theta}}Ff_{L_{i}}|^{+}_{-}(b_{i})|\leq\frac{1}{\sin\epsilon}A(N+1)\text{Re}(\tilde{\Omega}_{L_{0}}).$$

(47)

We are left with the contributions of $p_{0},p_{1},\ldots p_{N}$ to (38):

$$\text{Im}\sum_{0}^{N-1}e^{-i\hat{\theta}}F(f_{L_{j+1}}-f_{L_{j}})(p_{j}).$$

If we replace $f_{L_{i}}$ by its supremum value $\sup_{L_{i}}f_{L_{i}}$ for all $i=0,1,\ldots N$, the new expression would be

$$\begin{split}&\text{Im}\sum_{0}^{N-1}e^{-i\hat{\theta}}F(p_{j})(\sup f_{L_{j+1}}-\sup f_{L_{j}})\\ =&\text{Im}\sum_{1}^{N}e^{-i\hat{\theta}}\sup f_{L_{j}}(F(p_{j-1})-F(p_{j}))-\text{Im}(e^{-i\hat{\theta}}F(p)\sup f_{L_{0}})\end{split}$$

which is more intrinsically the integrand (45) of the elementary functional. Using the potential clustering assumption and the wedge region bound lemma, the error of replacing the potentials by $\sup_{L_{j}}f_{L_{j}}$ can be bounded by

$$2A\sum_{1}^{N}|F(p_{j})|\leq 2AN\frac{\text{Re}(F(p))}{\sin\epsilon}=\frac{2AN}{\sin\epsilon}\text{Re}(\tilde{\Omega}_{L_{0}}).$$

(48)

Now (46)(47)(48) are upper bounds on the three contributions to the difference between the Solomon functional integrand (39) and the elementary functional integrand. Their sum is bounded by

$$\frac{A(4N+2)}{\sin\epsilon}\text{Re}(\tilde{\Omega}_{L_{0}}),$$

so after integration on the moduli space,

$$|\mathcal{S}(L)-\bar{\mathcal{S}}(L)|\leq\frac{A(4N+2)}{\sin\epsilon}\int_{L_{0}}\text{Re }\Omega$$

as required. ∎