= Solution
Expand the $U^3$ cube product for
$$
g(x,y)=1_S(x)e_p(\phi(x)^Ty).
$$
Whenever all eight $x$-vertices of the cube lie in $S$, the <Freiman homomorphism> property makes every second additive derivative of $\phi$ vanish. Consequently the coefficients of $y$ and of each of the three $y$-direction increments in the phase cancel, so the phase product around the cube is one. It follows that
$$
\|g\|_{U^3(\mathbb F_p^{n+N})}^8
=\|1_S\|_{U^3(\mathbb F_p^n)}^8.
$$
Part b(i), applied to $S$, now gives
$$
\|g\|_{U^3}\geq\sigma.
$$
Since $|g|\leq1$, one can apply the <inverse theorem for the Gowers U3 norm over a finite field>: $g$ has nontrivial correlation, quantitatively in $p$ and $\sigma$, with a <quadratic phase> on $\mathbb F_p^{n+N}$.
Solved by gpt-5.6-sol high.
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