# Quartic SYK spectral edge SP-1-H0-to-H5

## 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 H5: For each even \(N\ge4\), let \(q_1,\ldots,q_N\) be self-adjoint matrices in an irreducible representation satisfying \(q_iq_j+q_jq_i=2\delta_{ij}I\). Put \(\Psi_S=-q_{i_1}q_{i_2}q_{i_3}q_{i_4}\) for \(S=\{i_1<i_2<i_3<i_4\}\), and let \(J_S\) be independent standard real Gaussians.
Define, using normalized trace,
\[
p_N(\beta)=\frac1N\log\mathbb E\operatorname{tr}
\exp\left(\beta\sqrt{\frac3{8N^3}}\sum_{|S|=4}J_S\Psi_S\right).
\]

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 \(\beta>0\), the sequence \(p_N(\beta)\), along even \(N\), has a finite limit \(p(\beta)\). For every pair \((\mu,\rho)\) satisfying the Dyson equation and \(\Sigma_\rho=\beta^2G_\mu^3\), that limit is
\[
p(\beta)=\sum_{n=0}^\infty\log\left(1+\int_0^\infty\frac{\rho(dx)}{((2n+1)\pi)^2+x^2}\right)
-\frac{3\beta^2}{8}\int_0^1G_\mu(u)^4\,du.
\]

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.