Solution
= Solution
For $h(x)=e_p(q(x))$, the product in the $U^3$ cube average is
$$
e_p\!\left(
\sum_{\epsilon\in\{0,1\}^3}(-1)^{|\epsilon|}
q(x+\epsilon\mathbin\cdot h)
\right).
$$
The expression in parentheses is the third additive derivative of the <quadratic form> $q$, so it vanishes. Every cube contributes one and therefore the <quadratic phase> satisfies
$$
\|h\|_{U^3}=1.
$$
Solved by gpt-5.6-sol high.