{
  "statement": "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\\]\n\nProve H6: 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\n\\[\nH_N=\\binom N4^{-1/2}\\sum_{|S|=4}J_S\\Psi_S.\n\\]\nThere exist a probability measure \\(\\rho_0\\) on \\([0,\\infty)\\) and a constant \\(\\kappa_4>0\\) such that, with\n\\[\ng_0(t)=\\frac12\\int_0^\\infty e^{-Et}\\,\\rho_0(dE),\n\\]\none has\n\\[\n\\int_0^\\infty E^2\\,\\rho_0(dE)=\\frac14,\\qquad g_0^3\\in L^1(0,\\infty),\n\\]\n\\[\n\\int_0^\\infty\\frac{v}{v^2+E^2}\\,\\rho_0(dE)\n=\\frac1{v+2\\int_0^\\infty\\sin(vt)g_0(t)^3\\,dt}\\qquad(v>0),\n\\]\nand\n\\[\n\\frac{\\lambda_{\\max}(H_N)}{\\sqrt N}\\longrightarrow\n\\kappa_4=4\\int_0^\\infty g_0(t)^4\\,dt\n\\quad\\text{almost surely as even }N\\to\\infty.\n\\]\nThe Gaussian families at different \\(N\\) may be coupled arbitrarily on a common probability space.",
  "starting_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\\]",
  "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\n\\[\nH_N=\\binom N4^{-1/2}\\sum_{|S|=4}J_S\\Psi_S.\n\\]\nThere exist a probability measure \\(\\rho_0\\) on \\([0,\\infty)\\) and a constant \\(\\kappa_4>0\\) such that, with\n\\[\ng_0(t)=\\frac12\\int_0^\\infty e^{-Et}\\,\\rho_0(dE),\n\\]\none has\n\\[\n\\int_0^\\infty E^2\\,\\rho_0(dE)=\\frac14,\\qquad g_0^3\\in L^1(0,\\infty),\n\\]\n\\[\n\\int_0^\\infty\\frac{v}{v^2+E^2}\\,\\rho_0(dE)\n=\\frac1{v+2\\int_0^\\infty\\sin(vt)g_0(t)^3\\,dt}\\qquad(v>0),\n\\]\nand\n\\[\n\\frac{\\lambda_{\\max}(H_N)}{\\sqrt N}\\longrightarrow\n\\kappa_4=4\\int_0^\\infty g_0(t)^4\\,dt\n\\quad\\text{almost surely as even }N\\to\\infty.\n\\]\nThe Gaussian families at different \\(N\\) may be coupled arbitrarily on a common probability space.",
  "case_id": "SP-1-H5-to-H6",
  "given_context": {
    "premise": "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\\]"
  },
  "protocol": "single"
}