First estimate. Set \(I_n^{(\beta )}=\int _{\mathcal{D}_\mathrm {core}}e^{-2t\| \lambda \| ^2}e^{\beta tQ}\, d\lambda \) with \(\beta =8\) (since \(|e^{4tQ}|^2=e^{8tQ}\)). By Lemma 16, \(Q_n(A)=Q_{n-1}(B)+\frac12 x^\top M(B)x -\frac{1}{12}\sum x_a^4\). Dropping the negative quartic and fixing \(B\), the \(x\)-fiber is \(\int _{\mathbb {R}^{n-1}}\exp (-2t\| x\| ^2+\frac{\beta t}{2}x^\top M(B)x)\, dx\). On \(\mathcal{D}_\mathrm {core}\), \(\| M(B)\| _{\mathrm{op}}\le 2s(B)\le 2d/t\), so \(I-\frac\beta 4 M(B)\) is positive definite for small \(d/t\). Lemma 10 evaluates the Gaussian integral; the determinant satisfies \(\det (I-\frac\beta 4 M(B))^{-1/2}\le \exp (C_\beta d^2/t^2)\) since \(\operatorname {tr}(M(B)^2)\le 4s(B)^2\) and \(\sum _j u_j=0\). Peeling off all \(n\) vertices gives \(I_n^{(\beta )}\le \exp (C_\beta nd^2/t^2)\, F(d,t)\), bounded by \(C_1G_{\mathrm{core}}(d,t)\) when \(nd^2/t^2\le c_1\) (using Lemma 11).

Second estimate. Use \(|e^u-1|^2\le u^2(1+e^{2u})\) with \(u=4tQ\). The term \(\int Q^2 e^{-2t\| \lambda \| ^2}\) equals \(F(d,t)\mathbb {E}_{\gamma _t}[Q^2]\), where \(\mathbb {E}_{\gamma _t}[Q^2]=O(d^2/t^4)\) by expanding \(Q=-\frac{1}{12}D+\frac18 C\) and using \(\mathbb {E}[DC]=0\) (parity) and \(\mathbb {E}[C^2]=O(d^2/t^4)\) (partner counting). The cross term \(\int Q^2 e^{8tQ}e^{-2t\| \lambda \| ^2}\) is bounded by Cauchy–Schwarz, the first estimate with \(\beta =16\), and hypercontractivity (Lemma 8) applied to the degree-4 polynomial \(Q\). Multiplying by \(16t^2\) gives the result.

By definition \(\kappa _5=\mathbb {E}[X_\lambda ^5]-10\mathbb {E}[X_\lambda ^3]\mathbb {E}[X_\lambda ^2]\), so the disconnected 5-edge multigraphs (triangle \(\times \) double edge) cancel. Thus \(P\) is supported on connected even 5-edge multigraphs. Such a graph has sum of degrees \(10\), minimum degree \(2\), hence at most \(5\) vertices: choosing \(\le 5\) vertices from \([n]\) gives \(O(n^5)\) monomials.

Under \(\gamma _t\) (variance \((4t)^{-1}\)), each monomial has degree 5. A pair \((M,M')\) survives \(\mathbb {E}_{\gamma _t}[MM']\) only if every edge has even total degree, i.e., \(\mathcal O(M')=\mathcal O(M)\) (same odd-support graph). The odd support is either a 5-cycle (unique monomial, unique partner) or a triangle on \(\{ i,j,k\} \) (partner types: \((3,1,1)\) on the triangle—\(3\) choices—or \((1,1,1,2)\) with a pendant edge to a new vertex \(\ell \)—\(O(n)\) choices). This gives \(O(n^5)\) surviving pairs. Each degree-10 moment is \(O(t^{-5})\); multiplying by \(F(d,t)\) and using Lemma 11 proves the claim.