{
  "statement": "Prove 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": null,
  "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-H0-to-H4",
  "given_context": {
    "premise": null
  },
  "protocol": "single"
}