# Quartic SYK spectral edge SP-1-H0-to-H4

## 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 H4: 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 \(x\ge0\) and \(0\le u\le1\), put
\[
k_x(u)=\frac{\cosh(x(1/2-u))}{2\cosh(x/2)},\qquad
G_\mu(u)=\int_0^\infty k_x(u)\,\mu(dx),\qquad
\Sigma_\rho(u)=\int_0^\infty k_x(u)\,\rho(dx),
\]
where \(\mu\) is a probability measure and \(\rho\) is a finite positive measure on \([0,\infty)\).

A pair \((\mu,\rho)\) satisfies the Dyson equation if
\[
\omega\int_0^\infty\frac{\mu(dx)}{\omega^2+x^2}
=\frac1{\omega+\omega\int_0^\infty\frac{\rho(dx)}{\omega^2+x^2}}\qquad(\omega>0).
\]

Put \(\Sigma^{(M)}(u)=\sum_{\alpha=1}^M v_\alpha^2k_{x_\alpha}(u)\) and
\[
G_{N,M,\theta}(u)=\frac1{2N}\sum_{i=1}^N
\frac{\mathbb E\operatorname{tr}(e^{(1-u)A_{N,M,\theta}}q_i e^{uA_{N,M,\theta}}q_i)}
{\mathbb E\operatorname{tr}e^{A_{N,M,\theta}}}.
\]
There exists a family \((\mu_\theta,\rho_\theta)_{\theta\in[0,1]}\) satisfying the Dyson equation and
\[
\Sigma_{\rho_\theta}(u)=(1-\theta)\Sigma^{(M)}(u)+\theta\beta^2G_{\mu_\theta}(u)^3
\qquad(0\le u\le1),
\]
such that, for every finite \(1\le r<\infty\),
\[
\lim_{\substack{N\to\infty\\N\ {\rm even}}}\sup_{\theta\in[0,1]}
\|G_{N,M,\theta}-G_{\mu_\theta}\|_{L^r(0,1)}=0.
\]

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.