# Quartic SYK spectral edge SP-1-H0-to-H3

## 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 H3: 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 probability measure \(\mu\) on \([0,\infty)\), write
\[
G_\mu(u)=\int_0^\infty\frac{\cosh(x(1/2-u))}{2\cosh(x/2)}\,\mu(dx),\qquad0\le u\le1.
\]

Let \(B_{N,M,\theta}\) be obtained from \(A_{N,M,\theta}\) by removing every random quartic term incident to site \(N\), while retaining all bath terms.
Let \(r\ge1\), let \(\mathfrak p\) be a pairing of \(\{1,\ldots,2r\}\), and assign a distinct bulk label \(i\in\{1,\ldots,N-1\}\) to each pair. Write \(j_a\) for the label of the pair containing position \(a\).
For \(0<t_1<\cdots<t_{2r}<1\), define
\[
W_N(\mathbf t,\mathfrak p,\mathbf i)=
\frac{\mathbb E\operatorname{tr}\left(
e^{t_1B}q_{j_1}e^{(t_2-t_1)B}q_{j_2}\cdots
e^{(t_{2r}-t_{2r-1})B}q_{j_{2r}}e^{(1-t_{2r})B}\right)}
{\mathbb E\operatorname{tr}e^B},\qquad B=B_{N,M,\theta}.
\]
Put \(\varepsilon(\mathfrak p)=(-1)^{\operatorname{cr}(\mathfrak p)}\), where \(\operatorname{cr}\) counts pairs of arcs with endpoints \(a<c<b<d\).
For every sequence of even \(N_j\to\infty\) and \(\theta_j\to\theta_\infty\in[0,1]\), there exist a subsequence and one Borel probability measure \(\Pi\) on \(\mathcal P([0,\infty))\), with its weak topology, such that simultaneously for all fixed \(r\) and \(\mathfrak p\),
\[
W_{N_j}(\mathbf t,\mathfrak p,\mathbf i)\longrightarrow
\varepsilon(\mathfrak p)\int_{\mathcal P([0,\infty))}
\prod_{\{a,b\}\in\mathfrak p,\ a<b}2G_\mu(t_b-t_a)\,\Pi(d\mu).
\]
Convergence is uniform over the distinct bulk labels and locally uniform in the open simplex \(0<t_1<\cdots<t_{2r}<1\). The same \(\Pi\) applies to every \(r\), pairing, and parameter choice.

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.