# Quartic SYK spectral edge SP-1-H1-to-H5

## 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 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.