{
  "statement": "H4: Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). There exist an integer \\(m\\ge2\\), a real number \\(k>0\\), a holomorphic function \\(\\phi\\) on a neighborhood of \\(\\overline D\\), and a real-valued function \\(U\\in C^2(\\overline D)\\), such that\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D),\n\\]\n\\[\n\\phi(e^{2\\pi i/m}z)=e^{2\\pi i/m}\\phi(z)\\quad(z\\in D),\n\\qquad \\phi\\not\\equiv z,\n\\]\nand\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.\n\nProve H5: There exist a bounded simply connected domain \\(\\Omega\\subset\\mathbb R^2\\) with \\(C^\\infty\\) boundary, a real number \\(k>0\\), and a nonzero real-valued function \\(u\\in C^2(\\overline\\Omega)\\), such that \\(\\Omega\\) is not a disk and\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
  "starting_statement": "Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). There exist an integer \\(m\\ge2\\), a real number \\(k>0\\), a holomorphic function \\(\\phi\\) on a neighborhood of \\(\\overline D\\), and a real-valued function \\(U\\in C^2(\\overline D)\\), such that\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D),\n\\]\n\\[\n\\phi(e^{2\\pi i/m}z)=e^{2\\pi i/m}\\phi(z)\\quad(z\\in D),\n\\qquad \\phi\\not\\equiv z,\n\\]\nand\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.",
  "target_statement": "There exist a bounded simply connected domain \\(\\Omega\\subset\\mathbb R^2\\) with \\(C^\\infty\\) boundary, a real number \\(k>0\\), and a nonzero real-valued function \\(u\\in C^2(\\overline\\Omega)\\), such that \\(\\Omega\\) is not a disk and\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
  "case_id": "EP-1-H4-to-H5",
  "given_context": {
    "premise": "Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). There exist an integer \\(m\\ge2\\), a real number \\(k>0\\), a holomorphic function \\(\\phi\\) on a neighborhood of \\(\\overline D\\), and a real-valued function \\(U\\in C^2(\\overline D)\\), such that\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D),\n\\]\n\\[\n\\phi(e^{2\\pi i/m}z)=e^{2\\pi i/m}\\phi(z)\\quad(z\\in D),\n\\qquad \\phi\\not\\equiv z,\n\\]\nand\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle."
  },
  "protocol": "single"
}