{
  "statement": "You may assume H2 below without proof. Prove H5 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: H5**\n\nFor a finite set \\(P\\subset\\mathbb R^2\\), let\n\\[\n\\nu(P)=\\#\\bigl\\{\\{x,y\\}\\subset P:x\\ne y,\\ \\|x-y\\|_2=1\\bigr\\},\n\\]\nwhere \\(\\|\\cdot\\|_2\\) is the Euclidean norm. For every positive integer \\(n\\), define\n\\[\n\\nu(n)=\\max\\{\\nu(P):P\\subset\\mathbb R^2,\\ |P|=n\\}.\n\\]\nThere exist a fixed real number \\(\\delta>0\\) and a sequence of finite planar sets \\((P_j)_{j\\ge1}\\) such that, writing \\(n_j=|P_j|\\),\n\\[\nn_j\\longrightarrow\\infty,\n\\qquad\n\\nu(P_j)\\ge n_j^{1+\\delta}\\quad\\text{for every }j\\ge1.\n\\]\nConsequently, infinitely many distinct positive integers \\(n\\) satisfy\n\\[\n\\nu(n)\\ge n^{1+\\delta}.\n\\]\nThe same \\(\\delta\\) applies to every set in the sequence.",
  "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": "For a finite set \\(P\\subset\\mathbb R^2\\), let\n\\[\n\\nu(P)=\\#\\bigl\\{\\{x,y\\}\\subset P:x\\ne y,\\ \\|x-y\\|_2=1\\bigr\\},\n\\]\nwhere \\(\\|\\cdot\\|_2\\) is the Euclidean norm. For every positive integer \\(n\\), define\n\\[\n\\nu(n)=\\max\\{\\nu(P):P\\subset\\mathbb R^2,\\ |P|=n\\}.\n\\]\nThere exist a fixed real number \\(\\delta>0\\) and a sequence of finite planar sets \\((P_j)_{j\\ge1}\\) such that, writing \\(n_j=|P_j|\\),\n\\[\nn_j\\longrightarrow\\infty,\n\\qquad\n\\nu(P_j)\\ge n_j^{1+\\delta}\\quad\\text{for every }j\\ge1.\n\\]\nConsequently, infinitely many distinct positive integers \\(n\\) satisfy\n\\[\n\\nu(n)\\ge n^{1+\\delta}.\n\\]\nThe same \\(\\delta\\) applies to every set in the sequence.",
  "case_id": "H2-to-H5",
  "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"
}