# Berenstein rigidity EP-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: For each integer \(m\ge2\) and each \(\rho>1\), define the following spaces and operators. Put \(D=\{z\in\mathbb C:|z|<1\}\). For \(\ell\in\mathbb Z\) and \(s\in\mathbb Z_{\ge0}\), set
\[
\Phi_{\ell,s}(r,\theta)=r^{m|\ell|}P_s^{(0,m|\ell|)}(2r^2-1)e^{im\ell\theta},
\]
where \(P_s^{(0,b)}\) is the Jacobi polynomial normalized by \(P_s^{(0,b)}(1)=1\).
Let \(W_{m,\rho}\) consist of complex coefficient series \(g=\sum_{\ell,s}g_{\ell,s}\Phi_{\ell,s}\) with
\[
\|g\|_\rho=\sum_{\ell,s}\rho^{|\ell|}|g_{\ell,s}|<\infty.
\]
Let \(Y_{m,\rho}\) be the real subspace of \(W_{m,\rho}\) consisting of real-valued functions invariant under \(\theta\mapsto-\theta\).
Let \(P_{m,\rho}\) consist of holomorphic series \(a(z)=\sum_{j\ge0}a_jz^{mj}\) with real coefficients and norm
\(\|a\|_P=\sum_{j\ge0}\rho^j|a_j|<\infty\).
Let \(B_{m,\rho}\) consist of real-valued, reflection-invariant boundary series
\(b(\theta)=\sum_{\ell\in\mathbb Z}b_\ell e^{im\ell\theta}\), with norm
\(\|b\|_B=\sum_\ell\rho^{|\ell|}|b_\ell|<\infty\).
For \(g\in Y_{m,\rho}\), let \(K_Dg\) denote the weak solution of \(\Delta U=g\) in \(D\) with zero Dirichlet trace; put \(Ng=\partial_r(K_Dg)|_{\partial D}\), with the derivative understood as its boundary trace.

For \(g\in Y_{m,\rho}\), \(k\in\mathbb R\), and \(a\in P_{m,\rho}\) with \(a_0=1\), define
\[
F_{m,\rho}(g,k,a)=\bigl(g+k^2|a|^2K_Dg,\ (Ng)^2-|a|^2|_{\partial D}\bigr).
\]
Its affine domain is
\[
X_{m,\rho}=\{(g,k,a)\in Y_{m,\rho}\times\mathbb R\times P_{m,\rho}:a_0=1\},
\]
with tangent space \(X_{m,\rho}^{\rm tan}=Y_{m,\rho}\times\mathbb R\times\{\dot a\in P_{m,\rho}:\dot a_0=0\}\) and norm
\[
\|(\dot g,\dot k,\dot a)\|_X=\|\dot g\|_\rho+|\dot k|+\|\dot a\|_P.
\]
Its codomain is \(Z_{m,\rho}=Y_{m,\rho}\times B_{m,\rho}\), with norm
\(\|(y,b)\|_Z=\|y\|_\rho+\|b\|_B\).

There exist an integer \(m\ge2\), a real number \(\rho>1\), a point \(x_c=(g_c,k_c,a_c)\in X_{m,\rho}\) with \(g_c\) and \(a_c\) of finite coefficient support, a bounded injective real-linear operator
\[
A:Z_{m,\rho}\longrightarrow X_{m,\rho}^{\rm tan},
\]
and real numbers \(r>0\) and \(0\le q<1\) with the following properties.
Write
\[
B_r(x_c)=\{x\in X_{m,\rho}:\|x-x_c\|_X\le r\}.
\]
The map \(F_{m,\rho}\) is well-defined and continuously Fréchet differentiable on an open neighborhood of \(B_r(x_c)\), and
\[
\|A F_{m,\rho}(x_c)\|_X<(1-q)r,
\qquad
\sup_{x\in B_r(x_c)}
\|I-A\,DF_{m,\rho}(x)\|_{X_{m,\rho}^{\rm tan}\to X_{m,\rho}^{\rm tan}}\le q,
\]
where \(I\) is the identity on \(X_{m,\rho}^{\rm tan}\).
For every \((g,k,a)\in B_r(x_c)\),
\[
k>0,\qquad
\operatorname{Re}a(z)>0\quad(z\in\overline D),\qquad
Ng(e^{i\theta})>0\quad(\theta\in\mathbb R),\qquad a_1\ne0.
\]

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.