inverse theorem\], we may assume that there are 1-bounded functions $g, h : [N] \to {\mathbb{C}}$ such that $$|\Lambda_{q, N}(g, h, f)|  {\geqslant}\delta .$$ Applying Lemma \[HB decomp\] to $g$ with ${\left\| \cdot\right\|} := {\left\| \cdot\right\|}_{q, N}^\flat$ and ${\varepsilon}:= {\tfrac{1}{2}}\delta N^{-1/2}$, we deduce that $$|\Lambda_{q, N}(g_{str}, h,f)|    {\geqslant}\delta  -  |\Lambda_{q, N}(g_{unf}, h,f)|
 {\geqslant}\delta -  {\left\| g_{unf}\right\|}_{q, N}^\flat {\geqslant}\tfrac{\delta}{2} .$$

Recalling that $M := \lfloor\sqrt{N/q}\rfloor$, define the dual function $$F(x) : = {\mathbb{E}}_{y \in [M]} h(x+y) f(x+qy^2).$$ The dual inequality then gives $$\tfrac\delta2  {\leqslant}|\Lambda_{q, N}(g_{str}, h,f)| = N^{-1}|{\left\langleg_{str}, F\right\rangle}| {\leqslant}2 \delta^{-1} {\left\| F\right\|}_{q, N}^\flat.$$ Hence, by the partial cut norm inverse theorem (Lemma \[partial inverse theorem\]), there exists a 1-bounded local function $\phi_1$ of resolution $\gg (\delta/q)^{O(1)}N^{1/2}$ and modulus $qq_1$ for some $q_1\ll  \delta^{-O(1)}$ such that $${\left| \sum_{x\in[N]} F(x)\phi_1(x)\right|} \gg \delta^{O(1)} N.$$ Thus $$|\Lambda_{q, N}(\phi_1, h, f)|  \gg \delta^{O(1)} .$$

We now re-run our argument, this time applying Lemma \[HB decomp\] to $h$ and deducing the existence of a 1-bounded local function $\phi_2$ of resolution $\gg (\delta/q)^{O(1)}N^{1/2}$ and modulus $qq_2$ for some $q_2\ll  \delta^{-O(1)}$ such that $$|\Lambda_{q, N}(\phi_1, \phi_2, f)|  \gg \delta^{O(1)} .$$ Expanding the counting operator and taking a maximum over $y\in [M]$ gives $$\begin{aligned}
\delta^{O(1)} NM & \ll {\left| 