{
  "statement": "You may assume H2 below without proof. Prove H4 below. No other checkpoint result is granted.\n\n**Given: H2**\n\nThere exist an absolute constant \\(C>0\\) and an integer \\(\\ell_0\\ge2\\) such that, for every integer \\(\\ell\\ge\\ell_0\\), putting\n\\[\nt=\\left\\lfloor\\frac{(\\ell-1)^2}{100}\\right\\rfloor,\n\\]\nthere exist a totally real cyclic cubic field \\(F/\\mathbb Q\\), distinct rational primes \\(q_1,\\ldots,q_t\\), and a tower\n\\[\nF=F_0\\subset F_1\\subset F_2\\subset\\cdots\n\\]\nsatisfying all the following properties.\n\n1. Every \\(q_b\\) is congruent to \\(1\\pmod4\\) and splits completely in every \\(F_j\\), for every \\(j\\ge0\\).\n\n2. Every \\(F_j/F\\) is finite Galois and everywhere unramified, and its Galois group is a \\(3\\)-group. Every \\(F_j\\) is totally real.\n\n3. Writing \\(f_j=[F_j:\\mathbb Q]\\), one has\n\\[\nf_j\\longrightarrow\\infty,\n\\qquad\n\\operatorname{rd}(F_j)=\\operatorname{rd}(F),\n\\qquad\n\\log\\operatorname{rd}(F)\\le C\\ell\\log\\ell.\n\\]\n\nFor each fixed \\(\\ell\\), the primes \\(q_1,\\ldots,q_t\\) are chosen once and are independent of \\(j\\).\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.\n\n**Target: H4**\n\nThere exist a fixed integer \\(m\\ge1\\), a fixed real number \\(\\gamma>0\\), a sequence of number fields \\((K_j)_{j\\ge1}\\), and finite sets \\(U_j\\subset K_j\\), such that, writing \\(d_j=[K_j:\\mathbb Q]\\), one has\n\\[\nd_j\\longrightarrow\\infty,\n\\qquad\nU_j\\subset m^{-1}\\mathcal O_{K_j},\n\\qquad\n|U_j|\\ge\\exp(\\gamma d_j).\n\\]\nFor every \\(j\\ge1\\), every \\(u\\in U_j\\), and every field embedding \\(\\sigma:K_j\\hookrightarrow\\mathbb C\\),\n\\[\n|\\sigma(u)|=1.\n\\]\n\nHere \\(\\mathcal O_{K_j}\\) is the ring of integers of \\(K_j\\), and \\(m^{-1}\\mathcal O_{K_j}=\\{a/m:a\\in\\mathcal O_{K_j}\\}\\). The same constants \\(m\\) and \\(\\gamma\\) apply to every \\(j\\).",
  "starting_statement": "There exist an absolute constant \\(C>0\\) and an integer \\(\\ell_0\\ge2\\) such that, for every integer \\(\\ell\\ge\\ell_0\\), putting\n\\[\nt=\\left\\lfloor\\frac{(\\ell-1)^2}{100}\\right\\rfloor,\n\\]\nthere exist a totally real cyclic cubic field \\(F/\\mathbb Q\\), distinct rational primes \\(q_1,\\ldots,q_t\\), and a tower\n\\[\nF=F_0\\subset F_1\\subset F_2\\subset\\cdots\n\\]\nsatisfying all the following properties.\n\n1. Every \\(q_b\\) is congruent to \\(1\\pmod4\\) and splits completely in every \\(F_j\\), for every \\(j\\ge0\\).\n\n2. Every \\(F_j/F\\) is finite Galois and everywhere unramified, and its Galois group is a \\(3\\)-group. Every \\(F_j\\) is totally real.\n\n3. Writing \\(f_j=[F_j:\\mathbb Q]\\), one has\n\\[\nf_j\\longrightarrow\\infty,\n\\qquad\n\\operatorname{rd}(F_j)=\\operatorname{rd}(F),\n\\qquad\n\\log\\operatorname{rd}(F)\\le C\\ell\\log\\ell.\n\\]\n\nFor each fixed \\(\\ell\\), the primes \\(q_1,\\ldots,q_t\\) are chosen once and are independent of \\(j\\).\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.",
  "target_statement": "There exist a fixed integer \\(m\\ge1\\), a fixed real number \\(\\gamma>0\\), a sequence of number fields \\((K_j)_{j\\ge1}\\), and finite sets \\(U_j\\subset K_j\\), such that, writing \\(d_j=[K_j:\\mathbb Q]\\), one has\n\\[\nd_j\\longrightarrow\\infty,\n\\qquad\nU_j\\subset m^{-1}\\mathcal O_{K_j},\n\\qquad\n|U_j|\\ge\\exp(\\gamma d_j).\n\\]\nFor every \\(j\\ge1\\), every \\(u\\in U_j\\), and every field embedding \\(\\sigma:K_j\\hookrightarrow\\mathbb C\\),\n\\[\n|\\sigma(u)|=1.\n\\]\n\nHere \\(\\mathcal O_{K_j}\\) is the ring of integers of \\(K_j\\), and \\(m^{-1}\\mathcal O_{K_j}=\\{a/m:a\\in\\mathcal O_{K_j}\\}\\). The same constants \\(m\\) and \\(\\gamma\\) apply to every \\(j\\).",
  "case_id": "H2-to-H4",
  "given_context": {
    "premise": "There exist an absolute constant \\(C>0\\) and an integer \\(\\ell_0\\ge2\\) such that, for every integer \\(\\ell\\ge\\ell_0\\), putting\n\\[\nt=\\left\\lfloor\\frac{(\\ell-1)^2}{100}\\right\\rfloor,\n\\]\nthere exist a totally real cyclic cubic field \\(F/\\mathbb Q\\), distinct rational primes \\(q_1,\\ldots,q_t\\), and a tower\n\\[\nF=F_0\\subset F_1\\subset F_2\\subset\\cdots\n\\]\nsatisfying all the following properties.\n\n1. Every \\(q_b\\) is congruent to \\(1\\pmod4\\) and splits completely in every \\(F_j\\), for every \\(j\\ge0\\).\n\n2. Every \\(F_j/F\\) is finite Galois and everywhere unramified, and its Galois group is a \\(3\\)-group. Every \\(F_j\\) is totally real.\n\n3. Writing \\(f_j=[F_j:\\mathbb Q]\\), one has\n\\[\nf_j\\longrightarrow\\infty,\n\\qquad\n\\operatorname{rd}(F_j)=\\operatorname{rd}(F),\n\\qquad\n\\log\\operatorname{rd}(F)\\le C\\ell\\log\\ell.\n\\]\n\nFor each fixed \\(\\ell\\), the primes \\(q_1,\\ldots,q_t\\) are chosen once and are independent of \\(j\\).\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."
  },
  "protocol": "single"
}