{
  "statement": "You may assume H4 below without proof. Prove H5 below. No other checkpoint result is granted.\n\n**Given: 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\\).\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 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\\).",
  "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": "H4-to-H5",
  "given_context": {
    "premise": "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\\)."
  },
  "protocol": "single"
}