{
  "statement": "Prove H5: For each even \\(N\\ge4\\), let \\(q_1,\\ldots,q_N\\) be self-adjoint matrices in an irreducible representation satisfying \\(q_iq_j+q_jq_i=2\\delta_{ij}I\\). Put \\(\\Psi_S=-q_{i_1}q_{i_2}q_{i_3}q_{i_4}\\) for \\(S=\\{i_1<i_2<i_3<i_4\\}\\), and let \\(J_S\\) be independent standard real Gaussians.\nDefine, using normalized trace,\n\\[\np_N(\\beta)=\\frac1N\\log\\mathbb E\\operatorname{tr}\n\\exp\\left(\\beta\\sqrt{\\frac3{8N^3}}\\sum_{|S|=4}J_S\\Psi_S\\right).\n\\]\n\nFor \\(x\\ge0\\) and \\(0\\le u\\le1\\), put\n\\[\nk_x(u)=\\frac{\\cosh(x(1/2-u))}{2\\cosh(x/2)},\\qquad\nG_\\mu(u)=\\int_0^\\infty k_x(u)\\,\\mu(dx),\\qquad\n\\Sigma_\\rho(u)=\\int_0^\\infty k_x(u)\\,\\rho(dx),\n\\]\nwhere \\(\\mu\\) is a probability measure and \\(\\rho\\) is a finite positive measure on \\([0,\\infty)\\).\n\nA pair \\((\\mu,\\rho)\\) satisfies the Dyson equation if\n\\[\n\\omega\\int_0^\\infty\\frac{\\mu(dx)}{\\omega^2+x^2}\n=\\frac1{\\omega+\\omega\\int_0^\\infty\\frac{\\rho(dx)}{\\omega^2+x^2}}\\qquad(\\omega>0).\n\\]\n\nFor every \\(\\beta>0\\), the sequence \\(p_N(\\beta)\\), along even \\(N\\), has a finite limit \\(p(\\beta)\\). For every pair \\((\\mu,\\rho)\\) satisfying the Dyson equation and \\(\\Sigma_\\rho=\\beta^2G_\\mu^3\\), that limit is\n\\[\np(\\beta)=\\sum_{n=0}^\\infty\\log\\left(1+\\int_0^\\infty\\frac{\\rho(dx)}{((2n+1)\\pi)^2+x^2}\\right)\n-\\frac{3\\beta^2}{8}\\int_0^1G_\\mu(u)^4\\,du.\n\\]",
  "starting_statement": null,
  "target_statement": "For each even \\(N\\ge4\\), let \\(q_1,\\ldots,q_N\\) be self-adjoint matrices in an irreducible representation satisfying \\(q_iq_j+q_jq_i=2\\delta_{ij}I\\). Put \\(\\Psi_S=-q_{i_1}q_{i_2}q_{i_3}q_{i_4}\\) for \\(S=\\{i_1<i_2<i_3<i_4\\}\\), and let \\(J_S\\) be independent standard real Gaussians.\nDefine, using normalized trace,\n\\[\np_N(\\beta)=\\frac1N\\log\\mathbb E\\operatorname{tr}\n\\exp\\left(\\beta\\sqrt{\\frac3{8N^3}}\\sum_{|S|=4}J_S\\Psi_S\\right).\n\\]\n\nFor \\(x\\ge0\\) and \\(0\\le u\\le1\\), put\n\\[\nk_x(u)=\\frac{\\cosh(x(1/2-u))}{2\\cosh(x/2)},\\qquad\nG_\\mu(u)=\\int_0^\\infty k_x(u)\\,\\mu(dx),\\qquad\n\\Sigma_\\rho(u)=\\int_0^\\infty k_x(u)\\,\\rho(dx),\n\\]\nwhere \\(\\mu\\) is a probability measure and \\(\\rho\\) is a finite positive measure on \\([0,\\infty)\\).\n\nA pair \\((\\mu,\\rho)\\) satisfies the Dyson equation if\n\\[\n\\omega\\int_0^\\infty\\frac{\\mu(dx)}{\\omega^2+x^2}\n=\\frac1{\\omega+\\omega\\int_0^\\infty\\frac{\\rho(dx)}{\\omega^2+x^2}}\\qquad(\\omega>0).\n\\]\n\nFor every \\(\\beta>0\\), the sequence \\(p_N(\\beta)\\), along even \\(N\\), has a finite limit \\(p(\\beta)\\). For every pair \\((\\mu,\\rho)\\) satisfying the Dyson equation and \\(\\Sigma_\\rho=\\beta^2G_\\mu^3\\), that limit is\n\\[\np(\\beta)=\\sum_{n=0}^\\infty\\log\\left(1+\\int_0^\\infty\\frac{\\rho(dx)}{((2n+1)\\pi)^2+x^2}\\right)\n-\\frac{3\\beta^2}{8}\\int_0^1G_\\mu(u)^4\\,du.\n\\]",
  "case_id": "SP-1-H0-to-H5",
  "given_context": {
    "premise": null
  },
  "protocol": "single"
}