{
  "statement": "H3: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nThe integral over the entire time axis is finite:\n\\[\n\\int_{\\mathbb R}\\|w(t)\\|_{L^2(\\mathbb T)}^2\\,dt<\\infty.\n\\]\n\nProve H4: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\). Suppose \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfies\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. If there exist \\(C\\in\\mathbb R\\), \\(T_0>0\\), and a nonempty open set \\(\\omega\\subset\\mathbb T\\) such that \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), then\n\\[\nu(x,t)=C\\qquad\\text{for every }(x,t)\\in\\mathbb T\\times\\mathbb R.\n\\]",
  "starting_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nThe integral over the entire time axis is finite:\n\\[\n\\int_{\\mathbb R}\\|w(t)\\|_{L^2(\\mathbb T)}^2\\,dt<\\infty.\n\\]",
  "target_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\). Suppose \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfies\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. If there exist \\(C\\in\\mathbb R\\), \\(T_0>0\\), and a nonempty open set \\(\\omega\\subset\\mathbb T\\) such that \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), then\n\\[\nu(x,t)=C\\qquad\\text{for every }(x,t)\\in\\mathbb T\\times\\mathbb R.\n\\]",
  "case_id": "NE-1-H3-to-H4",
  "given_context": {
    "premise": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nThe integral over the entire time axis is finite:\n\\[\n\\int_{\\mathbb R}\\|w(t)\\|_{L^2(\\mathbb T)}^2\\,dt<\\infty.\n\\]"
  },
  "protocol": "single"
}