# Quartic SYK spectral edge SP-1-H1-to-H4

## Setting

You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.

## Problem

H1: 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
\[
\omega\int_0^\infty\frac{\mu(dx)}{\omega^2+x^2}
=\frac1{\omega+\omega\int_0^\infty\frac{\rho(dx)}{\omega^2+x^2}}\qquad(\omega>0).
\]

For every \(\beta>0\), there is exactly one such pair satisfying
\[
\Sigma_\rho(u)=\beta^2G_\mu(u)^3\qquad(0\le u\le1).
\]

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.