{
  "statement": "Prove H1 below. No checkpoint result is granted.\n\n**Target: H1**\n\nThere exist an absolute constant \\(C>0\\) and an integer \\(\\ell_0\\ge2\\) such that, for every integer \\(\\ell\\ge\\ell_0\\), there exists a totally real cyclic cubic field \\(F/\\mathbb Q\\) with the following properties.\n\nLet \\(F^{\\mathrm{ur},3}\\) be the maximal everywhere-unramified pro-\\(3\\) extension of \\(F\\), and put \\(G=\\operatorname{Gal}(F^{\\mathrm{ur},3}/F)\\). Then\n\\[\nd(G)\\ge\\ell-1,\n\\qquad\n\\log\\operatorname{rd}(F)\\le C\\ell\\log\\ell.\n\\]\n\nHere \\(d(G)\\) is the minimum number of topological generators of \\(G\\).\n\nHere \\(\\operatorname{rd}(L)=|\\operatorname{Disc}(L)|^{1/[L:\\mathbb Q]}\\) is the root discriminant of a number field \\(L\\), and all logarithms are natural.",
  "starting_statement": null,
  "target_statement": "There exist an absolute constant \\(C>0\\) and an integer \\(\\ell_0\\ge2\\) such that, for every integer \\(\\ell\\ge\\ell_0\\), there exists a totally real cyclic cubic field \\(F/\\mathbb Q\\) with the following properties.\n\nLet \\(F^{\\mathrm{ur},3}\\) be the maximal everywhere-unramified pro-\\(3\\) extension of \\(F\\), and put \\(G=\\operatorname{Gal}(F^{\\mathrm{ur},3}/F)\\). Then\n\\[\nd(G)\\ge\\ell-1,\n\\qquad\n\\log\\operatorname{rd}(F)\\le C\\ell\\log\\ell.\n\\]\n\nHere \\(d(G)\\) is the minimum number of topological generators of \\(G\\).\n\nHere \\(\\operatorname{rd}(L)=|\\operatorname{Disc}(L)|^{1/[L:\\mathbb Q]}\\) is the root discriminant of a number field \\(L\\), and all logarithms are natural.",
  "case_id": "H0-to-H1",
  "given_context": {
    "premise": null
  },
  "protocol": "single"
}