# Quartic SYK spectral edge SP-1-H0-to-H2

## Setting

You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.

## Problem

Prove H2: Fix \(\beta>0\), \(M\ge1\), \(x_\alpha\ge0\), and \(v_\alpha\in\mathbb R\), with \(\sum_{\alpha=1}^M v_\alpha^2=\beta^2/4\).
For each even \(N\ge4\), let \(\chi_i,\eta_{i\alpha},\zeta_{i\alpha}\), \(1\le i\le N\), \(1\le\alpha\le M\), be self-adjoint Majorana operators in an irreducible representation: each squares to \(I/2\), and distinct operators anticommute. Put \(q_i=\sqrt2\chi_i\) and \(\Psi_S=-q_{i_1}q_{i_2}q_{i_3}q_{i_4}\) for \(S=\{i_1<i_2<i_3<i_4\}\).
Let \(J_S\) be independent standard real Gaussians, put \(c_N^2=3/(8N^3)\) with \(c_N>0\), and define
\[
A_{N,M,\theta}=\sqrt\theta\,\beta c_N\sum_{|S|=4}J_S\Psi_S
-\mathrm i\sum_{i,\alpha}\left(x_\alpha\eta_{i\alpha}\zeta_{i\alpha}
+\sqrt{1-\theta}\,v_\alpha\chi_i\eta_{i\alpha}\right),\qquad0\le\theta\le1.
\]
The trace \(\operatorname{tr}\) is normalized; expectation is over the Gaussians.

For a set \(I\subset\{1,\ldots,N\}\), let \(A^{\rm del}\) be obtained from \(A_{N,M,\theta}\) by omitting the random quartic terms with \(|S\cap I|\ge2\). For \(\sigma\in\{\pm1\}^I\), define
\[
A_\sigma^{\rm del}=\alpha_\sigma(A^{\rm del}),\qquad
\alpha_\sigma=\mathop{\prod}_{i\in I:\sigma_i=-1}\operatorname{Ad}(q_i),\qquad
\operatorname{Ad}(q_i)(B)=q_iBq_i,
\]
where the product is taken in increasing label order. If \(\sigma^{(i)}\) is obtained by reversing its \(i\)-th sign, set
\[
X_{i,\sigma}=\tfrac12(A_\sigma^{\rm del}-A_{\sigma^{(i)}}^{\rm del}),\qquad
Y_{i,\sigma}(t)=e^{-\mathrm itA_\sigma^{\rm del}}X_{i,\sigma}e^{\mathrm itA_\sigma^{\rm del}},
\]
\[
\Phi_\sigma(F)=\frac{\mathbb E\operatorname{tr}(e^{A_\sigma^{\rm del}}F)}{\mathbb E\operatorname{tr}e^{A_\sigma^{\rm del}}}.
\]
Let \(\mathfrak H_{m,T}\) consist of products \(L=\prod_{\ell=1}^s e^{\mathrm it_\ell A_{\tau_\ell}^{\rm del}}\), where \(0\le s\le m\), \(\tau_\ell\in\{\pm1\}^I\), and \(\sum_\ell|t_\ell|\le T\).
For every fixed integer \(k\ge2\), integer \(m\ge0\), and finite \(T,R\ge0\),
\[
\lim_{\substack{N\to\infty\\N\ {\rm even}}}
\sup_{\substack{|I|=k,\ i,j\in I,\ i\ne j\\
\theta\in[0,1],\ \sigma\in\{\pm1\}^I,\ L\in\mathfrak H_{m,T}\\|s|,|t|\le R}}
\Phi_\sigma\left(L C_{ij,\sigma}(s,t)C_{ij,\sigma}(s,t)^*L^*\right)=0,
\]
where \(C_{ij,\sigma}(s,t)=[Y_{i,\sigma}(s),Y_{j,\sigma}(t)]\).

Write the full proof to answer.md in the workspace root. Write the statements and hypotheses of any external results you cite to cited_results.md; if none are used, write 'No external results cited.' Identify any gap you cannot close.