4.3 Log-expansion and small-ball decay
There are \(c_2,C_2{\gt}0\) such that whenever \(s(\lambda )\le c_2\),
with \(|E_6(\lambda )|\le C_2 s(\lambda )^3\).
Set \(K(u)=\log \mathbb {E}[e^{iuX_\lambda }]\) and Taylor-expand to fifth order. Lemma 6 ensures \(\operatorname {Re}\psi (u\lambda )\ge 3/4\), so the principal branch of \(\log \) is well-defined on \([0,1]\). The sixth derivative \(K^{(6)}\) involves products \(m(u)^{-k}\prod m^{(j_\ell )}(u)\); each factor is bounded using \(|m(u)|\ge 3/4\) and hypercontractivity (Lemma 8) applied to the degree-2 polynomial \(X_\lambda \).
There exists \(r_0\in (0,\pi /4)\) such that \(|\psi (\lambda )|^{4t}\le e^{-3t\| \lambda \| ^2/2}\) for every \(t\ge 1\) and \(\lambda \in \mathcal{D}_{r_0}\).
By the log-expansion (Lemma 19), \(\operatorname {Re}\log \psi =-\frac12 s+Q+\operatorname {Re}E_6\). Show \(Q=O(s^2)\) using \(|\mathcal{C}_4|\le \operatorname {tr}(A^2)^2=4s^2\) and \(|E_6|\le C_2 s^3\). For small \(r_0\), the corrections are absorbed by the leading \(-\frac12 s\) term.