4.4 Cubic phase bound
For all \(d,t\ge 1\),
\[ \operatorname {Re}\int _{\mathcal{D}_\mathrm {core}}e^{-2t\| \lambda \| ^2}e^{-4itT(\lambda )}\, d\lambda \ge \bigl[1-O(n^3/t)\bigr]G_{\mathrm{core}}(d,t). \]
Proof
Since \(T\) is odd and \(\mathcal{D}_\mathrm {core}\) is symmetric, the imaginary part vanishes. The real part satisfies \(1-\cos (4tT)\le 8t^2 T^2\). By the triangle formula (Lemma 14), \(T^2\) is a degree-6 polynomial. Gaussian radial moments (Lemma 9) bound the integral of \(T^2\) against the Gaussian weight, giving an \(O(n^3/t)\) correction.