{
  "H0-to-H1": {
    "id": "H0-to-H1",
    "eval_version": "v5",
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "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.",
    "case_sha256": "54fe71b5ccdb300ca98761deed2ee50aea71b581ea6d2f5d6a956a7242eddfb3",
    "checkpoints_sha256": "3c4613421ae2f328b55d76d720788d1f54dfd0121b9883bc2843600395303559",
    "protocol_sha256": "0d2ea2d385c8f439916f7604d96d6d877b3ea6042832aa75e4fe3f56755c9604"
  },
  "H0-to-H2": {
    "id": "H0-to-H2",
    "eval_version": "v5",
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H2 below. No checkpoint result is granted.\n\n**Target: 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.",
    "case_sha256": "34a1787991fe17a77513079539adc210f57a9012ed87c045f4d0f65cf5bd6ae9",
    "checkpoints_sha256": "3c4613421ae2f328b55d76d720788d1f54dfd0121b9883bc2843600395303559",
    "protocol_sha256": "0d2ea2d385c8f439916f7604d96d6d877b3ea6042832aa75e4fe3f56755c9604"
  },
  "H0-to-H3": {
    "id": "H0-to-H3",
    "eval_version": "v5",
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H3 below. No checkpoint result is granted.\n\n**Target: H3**\n\nThere exists an absolute constant \\(C\\ge1\\) such that, for every totally real number field \\(F\\) of degree \\(f=[F:\\mathbb Q]\\), the field\n\\[\nK=F(\\mathrm i),\\qquad \\mathrm i^2=-1,\n\\]\nsatisfies\n\\[\n\\operatorname{rd}(K)\\le2\\operatorname{rd}(F),\n\\qquad\nh(K)\\le\\bigl(C\\operatorname{rd}(F)\\bigr)^{2f}.\n\\]\n\nHere \\(h(K)\\) is the ideal class number of the ring of integers of \\(K\\). The constant \\(C\\) is independent of \\(F\\) and \\(f\\).\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_sha256": "0694237f18f666450ad39c208aee010e125b6f9e24fec2c3ee8a405e4829ab83",
    "checkpoints_sha256": "3c4613421ae2f328b55d76d720788d1f54dfd0121b9883bc2843600395303559",
    "protocol_sha256": "0d2ea2d385c8f439916f7604d96d6d877b3ea6042832aa75e4fe3f56755c9604"
  },
  "H0-to-H4": {
    "id": "H0-to-H4",
    "eval_version": "v5",
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H4 below. No checkpoint result is granted.\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\\).",
    "case_sha256": "d119f72538fbce7a6291931e2de135c96f57d9b251766774debc7a97708aef16",
    "checkpoints_sha256": "3c4613421ae2f328b55d76d720788d1f54dfd0121b9883bc2843600395303559",
    "protocol_sha256": "0d2ea2d385c8f439916f7604d96d6d877b3ea6042832aa75e4fe3f56755c9604"
  },
  "H0-to-H5": {
    "id": "H0-to-H5",
    "eval_version": "v5",
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H5 below. No checkpoint result is granted.\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.",
    "case_sha256": "1b0e80ad100154560fe0f95aacd8c9506d1f0677f90d2420b4240d2b584149da",
    "checkpoints_sha256": "3c4613421ae2f328b55d76d720788d1f54dfd0121b9883bc2843600395303559",
    "protocol_sha256": "0d2ea2d385c8f439916f7604d96d6d877b3ea6042832aa75e4fe3f56755c9604"
  },
  "H1-to-H2": {
    "id": "H1-to-H2",
    "eval_version": "v5",
    "system": "You are solving a mathematical proof problem. Assume the stated starting result and establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "You may assume H1 below without proof. Prove H2 below. No other checkpoint result is granted.\n\n**Given: 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.\n\n**Target: 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.",
    "case_sha256": "36a56b3d42be7c161ff5e9acb3324b8cb66a0a59351c6c452b607bac89434d07",
    "checkpoints_sha256": "3c4613421ae2f328b55d76d720788d1f54dfd0121b9883bc2843600395303559",
    "protocol_sha256": "0d2ea2d385c8f439916f7604d96d6d877b3ea6042832aa75e4fe3f56755c9604"
  },
  "H1-to-H3": {
    "id": "H1-to-H3",
    "eval_version": "v5",
    "system": "You are solving a mathematical proof problem. Assume the stated starting result and establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "You may assume H1 below without proof. Prove H3 below. No other checkpoint result is granted.\n\n**Given: 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.\n\n**Target: H3**\n\nThere exists an absolute constant \\(C\\ge1\\) such that, for every totally real number field \\(F\\) of degree \\(f=[F:\\mathbb Q]\\), the field\n\\[\nK=F(\\mathrm i),\\qquad \\mathrm i^2=-1,\n\\]\nsatisfies\n\\[\n\\operatorname{rd}(K)\\le2\\operatorname{rd}(F),\n\\qquad\nh(K)\\le\\bigl(C\\operatorname{rd}(F)\\bigr)^{2f}.\n\\]\n\nHere \\(h(K)\\) is the ideal class number of the ring of integers of \\(K\\). The constant \\(C\\) is independent of \\(F\\) and \\(f\\).\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_sha256": "c5271a19eae79986e2c0abf49a3b3233535d82ee0c158a38732f4c786156b015",
    "checkpoints_sha256": "3c4613421ae2f328b55d76d720788d1f54dfd0121b9883bc2843600395303559",
    "protocol_sha256": "0d2ea2d385c8f439916f7604d96d6d877b3ea6042832aa75e4fe3f56755c9604"
  },
  "H1-to-H4": {
    "id": "H1-to-H4",
    "eval_version": "v5",
    "system": "You are solving a mathematical proof problem. Assume the stated starting result and establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "You may assume H1 below without proof. Prove H4 below. No other checkpoint result is granted.\n\n**Given: 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.\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\\).",
    "case_sha256": "b4afe831d754934a1c642e9d6c3e1a3cda033952a1de76f153d7715336e2d2fc",
    "checkpoints_sha256": "3c4613421ae2f328b55d76d720788d1f54dfd0121b9883bc2843600395303559",
    "protocol_sha256": "0d2ea2d385c8f439916f7604d96d6d877b3ea6042832aa75e4fe3f56755c9604"
  },
  "H1-to-H5": {
    "id": "H1-to-H5",
    "eval_version": "v5",
    "system": "You are solving a mathematical proof problem. Assume the stated starting result and establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "You may assume H1 below without proof. Prove H5 below. No other checkpoint result is granted.\n\n**Given: 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.\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.",
    "case_sha256": "65d4fab3979cbdfd258498edc2c0801a2f87aef65f6558640cd3658cdf0a6857",
    "checkpoints_sha256": "3c4613421ae2f328b55d76d720788d1f54dfd0121b9883bc2843600395303559",
    "protocol_sha256": "0d2ea2d385c8f439916f7604d96d6d877b3ea6042832aa75e4fe3f56755c9604"
  },
  "H2-to-H3": {
    "id": "H2-to-H3",
    "eval_version": "v5",
    "system": "You are solving a mathematical proof problem. Assume the stated starting result and establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "You may assume H2 below without proof. Prove H3 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: H3**\n\nThere exists an absolute constant \\(C\\ge1\\) such that, for every totally real number field \\(F\\) of degree \\(f=[F:\\mathbb Q]\\), the field\n\\[\nK=F(\\mathrm i),\\qquad \\mathrm i^2=-1,\n\\]\nsatisfies\n\\[\n\\operatorname{rd}(K)\\le2\\operatorname{rd}(F),\n\\qquad\nh(K)\\le\\bigl(C\\operatorname{rd}(F)\\bigr)^{2f}.\n\\]\n\nHere \\(h(K)\\) is the ideal class number of the ring of integers of \\(K\\). The constant \\(C\\) is independent of \\(F\\) and \\(f\\).\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_sha256": "3fb78854edb4a8a4441a2b29a1743490292e2399f37bf3002c5f7666e7595f8d",
    "checkpoints_sha256": "3c4613421ae2f328b55d76d720788d1f54dfd0121b9883bc2843600395303559",
    "protocol_sha256": "0d2ea2d385c8f439916f7604d96d6d877b3ea6042832aa75e4fe3f56755c9604"
  },
  "H2-to-H4": {
    "id": "H2-to-H4",
    "eval_version": "v5",
    "system": "You are solving a mathematical proof problem. Assume the stated starting result and establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "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\\).",
    "case_sha256": "91c5947c700a89af76b897d67a715ef12763d2fe3cf38a47f46e06e83f76a8ce",
    "checkpoints_sha256": "3c4613421ae2f328b55d76d720788d1f54dfd0121b9883bc2843600395303559",
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  },
  "H2-to-H5": {
    "id": "H2-to-H5",
    "eval_version": "v5",
    "system": "You are solving a mathematical proof problem. Assume the stated starting result and establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "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.",
    "case_sha256": "1d060cb974e43d996f7c75278f8294edc2d9934c71204b1ddbe1c341398a5a5b",
    "checkpoints_sha256": "3c4613421ae2f328b55d76d720788d1f54dfd0121b9883bc2843600395303559",
    "protocol_sha256": "0d2ea2d385c8f439916f7604d96d6d877b3ea6042832aa75e4fe3f56755c9604"
  },
  "H3-to-H4": {
    "id": "H3-to-H4",
    "eval_version": "v5",
    "system": "You are solving a mathematical proof problem. Assume the stated starting result and establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "You may assume H3 below without proof. Prove H4 below. No other checkpoint result is granted.\n\n**Given: H3**\n\nThere exists an absolute constant \\(C\\ge1\\) such that, for every totally real number field \\(F\\) of degree \\(f=[F:\\mathbb Q]\\), the field\n\\[\nK=F(\\mathrm i),\\qquad \\mathrm i^2=-1,\n\\]\nsatisfies\n\\[\n\\operatorname{rd}(K)\\le2\\operatorname{rd}(F),\n\\qquad\nh(K)\\le\\bigl(C\\operatorname{rd}(F)\\bigr)^{2f}.\n\\]\n\nHere \\(h(K)\\) is the ideal class number of the ring of integers of \\(K\\). The constant \\(C\\) is independent of \\(F\\) and \\(f\\).\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\\).",
    "case_sha256": "9c572864ef0f22e69477d2c7a31b908900e088c68a1035f2953502875fc673c1",
    "checkpoints_sha256": "3c4613421ae2f328b55d76d720788d1f54dfd0121b9883bc2843600395303559",
    "protocol_sha256": "0d2ea2d385c8f439916f7604d96d6d877b3ea6042832aa75e4fe3f56755c9604"
  },
  "H3-to-H5": {
    "id": "H3-to-H5",
    "eval_version": "v5",
    "system": "You are solving a mathematical proof problem. Assume the stated starting result and establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "You may assume H3 below without proof. Prove H5 below. No other checkpoint result is granted.\n\n**Given: H3**\n\nThere exists an absolute constant \\(C\\ge1\\) such that, for every totally real number field \\(F\\) of degree \\(f=[F:\\mathbb Q]\\), the field\n\\[\nK=F(\\mathrm i),\\qquad \\mathrm i^2=-1,\n\\]\nsatisfies\n\\[\n\\operatorname{rd}(K)\\le2\\operatorname{rd}(F),\n\\qquad\nh(K)\\le\\bigl(C\\operatorname{rd}(F)\\bigr)^{2f}.\n\\]\n\nHere \\(h(K)\\) is the ideal class number of the ring of integers of \\(K\\). The constant \\(C\\) is independent of \\(F\\) and \\(f\\).\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.",
    "case_sha256": "96d1a002adfa0f22c57e5b64a7a96e6b2a03a9ab178b17a2d0c69d9a062954de",
    "checkpoints_sha256": "3c4613421ae2f328b55d76d720788d1f54dfd0121b9883bc2843600395303559",
    "protocol_sha256": "0d2ea2d385c8f439916f7604d96d6d877b3ea6042832aa75e4fe3f56755c9604"
  },
  "H4-to-H5": {
    "id": "H4-to-H5",
    "eval_version": "v5",
    "system": "You are solving a mathematical proof problem. Assume the stated starting result and establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "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.",
    "case_sha256": "3c17ce03d659538df8597468d23fa19c0f80bc8b5d80a6e5685c9cef0976bdba",
    "checkpoints_sha256": "3c4613421ae2f328b55d76d720788d1f54dfd0121b9883bc2843600395303559",
    "protocol_sha256": "0d2ea2d385c8f439916f7604d96d6d877b3ea6042832aa75e4fe3f56755c9604"
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}