{
  "statement": "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\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(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.\n\nFor \\(g\\in Y_{m,\\rho}\\), \\(k\\in\\mathbb R\\), and \\(a\\in P_{m,\\rho}\\) with \\(a_0=1\\), define\n\\[\nF_{m,\\rho}(g,k,a)=\\bigl(g+k^2|a|^2K_Dg,\\ (Ng)^2-|a|^2|_{\\partial D}\\bigr).\n\\]\nIts affine domain is\n\\[\nX_{m,\\rho}=\\{(g,k,a)\\in Y_{m,\\rho}\\times\\mathbb R\\times P_{m,\\rho}:a_0=1\\},\n\\]\nwith 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\n\\[\n\\|(\\dot g,\\dot k,\\dot a)\\|_X=\\|\\dot g\\|_\\rho+|\\dot k|+\\|\\dot a\\|_P.\n\\]\nIts codomain is \\(Z_{m,\\rho}=Y_{m,\\rho}\\times B_{m,\\rho}\\), with norm\n\\(\\|(y,b)\\|_Z=\\|y\\|_\\rho+\\|b\\|_B\\).\n\nThere 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\n\\[\nA:Z_{m,\\rho}\\longrightarrow X_{m,\\rho}^{\\rm tan},\n\\]\nand real numbers \\(r>0\\) and \\(0\\le q<1\\) with the following properties.\nWrite\n\\[\nB_r(x_c)=\\{x\\in X_{m,\\rho}:\\|x-x_c\\|_X\\le r\\}.\n\\]\nThe map \\(F_{m,\\rho}\\) is well-defined and continuously Fréchet differentiable on an open neighborhood of \\(B_r(x_c)\\), and\n\\[\n\\|A F_{m,\\rho}(x_c)\\|_X<(1-q)r,\n\\qquad\n\\sup_{x\\in B_r(x_c)}\n\\|I-A\\,DF_{m,\\rho}(x)\\|_{X_{m,\\rho}^{\\rm tan}\\to X_{m,\\rho}^{\\rm tan}}\\le q,\n\\]\nwhere \\(I\\) is the identity on \\(X_{m,\\rho}^{\\rm tan}\\).\nFor every \\((g,k,a)\\in B_r(x_c)\\),\n\\[\nk>0,\\qquad\n\\operatorname{Re}a(z)>0\\quad(z\\in\\overline D),\\qquad\nNg(e^{i\\theta})>0\\quad(\\theta\\in\\mathbb R),\\qquad a_1\\ne0.\n\\]",
  "starting_statement": null,
  "target_statement": "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\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(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.\n\nFor \\(g\\in Y_{m,\\rho}\\), \\(k\\in\\mathbb R\\), and \\(a\\in P_{m,\\rho}\\) with \\(a_0=1\\), define\n\\[\nF_{m,\\rho}(g,k,a)=\\bigl(g+k^2|a|^2K_Dg,\\ (Ng)^2-|a|^2|_{\\partial D}\\bigr).\n\\]\nIts affine domain is\n\\[\nX_{m,\\rho}=\\{(g,k,a)\\in Y_{m,\\rho}\\times\\mathbb R\\times P_{m,\\rho}:a_0=1\\},\n\\]\nwith 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\n\\[\n\\|(\\dot g,\\dot k,\\dot a)\\|_X=\\|\\dot g\\|_\\rho+|\\dot k|+\\|\\dot a\\|_P.\n\\]\nIts codomain is \\(Z_{m,\\rho}=Y_{m,\\rho}\\times B_{m,\\rho}\\), with norm\n\\(\\|(y,b)\\|_Z=\\|y\\|_\\rho+\\|b\\|_B\\).\n\nThere 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\n\\[\nA:Z_{m,\\rho}\\longrightarrow X_{m,\\rho}^{\\rm tan},\n\\]\nand real numbers \\(r>0\\) and \\(0\\le q<1\\) with the following properties.\nWrite\n\\[\nB_r(x_c)=\\{x\\in X_{m,\\rho}:\\|x-x_c\\|_X\\le r\\}.\n\\]\nThe map \\(F_{m,\\rho}\\) is well-defined and continuously Fréchet differentiable on an open neighborhood of \\(B_r(x_c)\\), and\n\\[\n\\|A F_{m,\\rho}(x_c)\\|_X<(1-q)r,\n\\qquad\n\\sup_{x\\in B_r(x_c)}\n\\|I-A\\,DF_{m,\\rho}(x)\\|_{X_{m,\\rho}^{\\rm tan}\\to X_{m,\\rho}^{\\rm tan}}\\le q,\n\\]\nwhere \\(I\\) is the identity on \\(X_{m,\\rho}^{\\rm tan}\\).\nFor every \\((g,k,a)\\in B_r(x_c)\\),\n\\[\nk>0,\\qquad\n\\operatorname{Re}a(z)>0\\quad(z\\in\\overline D),\\qquad\nNg(e^{i\\theta})>0\\quad(\\theta\\in\\mathbb R),\\qquad a_1\\ne0.\n\\]",
  "case_id": "EP-1-H0-to-H3",
  "given_context": {
    "premise": null
  },
  "protocol": "single"
}