{
  "statement": "H4: Fix \\(\\beta>0\\), \\(M\\ge1\\), \\(x_\\alpha\\ge0\\), and \\(v_\\alpha\\in\\mathbb R\\), with \\(\\sum_{\\alpha=1}^M v_\\alpha^2=\\beta^2/4\\).\nFor each even \\(N\\ge4\\), let \\(\\chi_i,\\eta_{i\\alpha},\\zeta_{i\\alpha}\\), \\(1\\le i\\le N\\), \\(1\\le\\alpha\\le M\\), be self-adjoint Majorana operators in an irreducible representation: each squares to \\(I/2\\), and distinct operators anticommute. Put \\(q_i=\\sqrt2\\chi_i\\) and \\(\\Psi_S=-q_{i_1}q_{i_2}q_{i_3}q_{i_4}\\) for \\(S=\\{i_1<i_2<i_3<i_4\\}\\).\nLet \\(J_S\\) be independent standard real Gaussians, put \\(c_N^2=3/(8N^3)\\) with \\(c_N>0\\), and define\n\\[\nA_{N,M,\\theta}=\\sqrt\\theta\\,\\beta c_N\\sum_{|S|=4}J_S\\Psi_S\n-\\mathrm i\\sum_{i,\\alpha}\\left(x_\\alpha\\eta_{i\\alpha}\\zeta_{i\\alpha}\n+\\sqrt{1-\\theta}\\,v_\\alpha\\chi_i\\eta_{i\\alpha}\\right),\\qquad0\\le\\theta\\le1.\n\\]\nThe trace \\(\\operatorname{tr}\\) is normalized; expectation is over the Gaussians.\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\nPut \\(\\Sigma^{(M)}(u)=\\sum_{\\alpha=1}^M v_\\alpha^2k_{x_\\alpha}(u)\\) and\n\\[\nG_{N,M,\\theta}(u)=\\frac1{2N}\\sum_{i=1}^N\n\\frac{\\mathbb E\\operatorname{tr}(e^{(1-u)A_{N,M,\\theta}}q_i e^{uA_{N,M,\\theta}}q_i)}\n{\\mathbb E\\operatorname{tr}e^{A_{N,M,\\theta}}}.\n\\]\nThere exists a family \\((\\mu_\\theta,\\rho_\\theta)_{\\theta\\in[0,1]}\\) satisfying the Dyson equation and\n\\[\n\\Sigma_{\\rho_\\theta}(u)=(1-\\theta)\\Sigma^{(M)}(u)+\\theta\\beta^2G_{\\mu_\\theta}(u)^3\n\\qquad(0\\le u\\le1),\n\\]\nsuch that, for every finite \\(1\\le r<\\infty\\),\n\\[\n\\lim_{\\substack{N\\to\\infty\\\\N\\ {\\rm even}}}\\sup_{\\theta\\in[0,1]}\n\\|G_{N,M,\\theta}-G_{\\mu_\\theta}\\|_{L^r(0,1)}=0.\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": "Fix \\(\\beta>0\\), \\(M\\ge1\\), \\(x_\\alpha\\ge0\\), and \\(v_\\alpha\\in\\mathbb R\\), with \\(\\sum_{\\alpha=1}^M v_\\alpha^2=\\beta^2/4\\).\nFor each even \\(N\\ge4\\), let \\(\\chi_i,\\eta_{i\\alpha},\\zeta_{i\\alpha}\\), \\(1\\le i\\le N\\), \\(1\\le\\alpha\\le M\\), be self-adjoint Majorana operators in an irreducible representation: each squares to \\(I/2\\), and distinct operators anticommute. Put \\(q_i=\\sqrt2\\chi_i\\) and \\(\\Psi_S=-q_{i_1}q_{i_2}q_{i_3}q_{i_4}\\) for \\(S=\\{i_1<i_2<i_3<i_4\\}\\).\nLet \\(J_S\\) be independent standard real Gaussians, put \\(c_N^2=3/(8N^3)\\) with \\(c_N>0\\), and define\n\\[\nA_{N,M,\\theta}=\\sqrt\\theta\\,\\beta c_N\\sum_{|S|=4}J_S\\Psi_S\n-\\mathrm i\\sum_{i,\\alpha}\\left(x_\\alpha\\eta_{i\\alpha}\\zeta_{i\\alpha}\n+\\sqrt{1-\\theta}\\,v_\\alpha\\chi_i\\eta_{i\\alpha}\\right),\\qquad0\\le\\theta\\le1.\n\\]\nThe trace \\(\\operatorname{tr}\\) is normalized; expectation is over the Gaussians.\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\nPut \\(\\Sigma^{(M)}(u)=\\sum_{\\alpha=1}^M v_\\alpha^2k_{x_\\alpha}(u)\\) and\n\\[\nG_{N,M,\\theta}(u)=\\frac1{2N}\\sum_{i=1}^N\n\\frac{\\mathbb E\\operatorname{tr}(e^{(1-u)A_{N,M,\\theta}}q_i e^{uA_{N,M,\\theta}}q_i)}\n{\\mathbb E\\operatorname{tr}e^{A_{N,M,\\theta}}}.\n\\]\nThere exists a family \\((\\mu_\\theta,\\rho_\\theta)_{\\theta\\in[0,1]}\\) satisfying the Dyson equation and\n\\[\n\\Sigma_{\\rho_\\theta}(u)=(1-\\theta)\\Sigma^{(M)}(u)+\\theta\\beta^2G_{\\mu_\\theta}(u)^3\n\\qquad(0\\le u\\le1),\n\\]\nsuch that, for every finite \\(1\\le r<\\infty\\),\n\\[\n\\lim_{\\substack{N\\to\\infty\\\\N\\ {\\rm even}}}\\sup_{\\theta\\in[0,1]}\n\\|G_{N,M,\\theta}-G_{\\mu_\\theta}\\|_{L^r(0,1)}=0.\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-H4-to-H6",
  "given_context": {
    "premise": "Fix \\(\\beta>0\\), \\(M\\ge1\\), \\(x_\\alpha\\ge0\\), and \\(v_\\alpha\\in\\mathbb R\\), with \\(\\sum_{\\alpha=1}^M v_\\alpha^2=\\beta^2/4\\).\nFor each even \\(N\\ge4\\), let \\(\\chi_i,\\eta_{i\\alpha},\\zeta_{i\\alpha}\\), \\(1\\le i\\le N\\), \\(1\\le\\alpha\\le M\\), be self-adjoint Majorana operators in an irreducible representation: each squares to \\(I/2\\), and distinct operators anticommute. Put \\(q_i=\\sqrt2\\chi_i\\) and \\(\\Psi_S=-q_{i_1}q_{i_2}q_{i_3}q_{i_4}\\) for \\(S=\\{i_1<i_2<i_3<i_4\\}\\).\nLet \\(J_S\\) be independent standard real Gaussians, put \\(c_N^2=3/(8N^3)\\) with \\(c_N>0\\), and define\n\\[\nA_{N,M,\\theta}=\\sqrt\\theta\\,\\beta c_N\\sum_{|S|=4}J_S\\Psi_S\n-\\mathrm i\\sum_{i,\\alpha}\\left(x_\\alpha\\eta_{i\\alpha}\\zeta_{i\\alpha}\n+\\sqrt{1-\\theta}\\,v_\\alpha\\chi_i\\eta_{i\\alpha}\\right),\\qquad0\\le\\theta\\le1.\n\\]\nThe trace \\(\\operatorname{tr}\\) is normalized; expectation is over the Gaussians.\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\nPut \\(\\Sigma^{(M)}(u)=\\sum_{\\alpha=1}^M v_\\alpha^2k_{x_\\alpha}(u)\\) and\n\\[\nG_{N,M,\\theta}(u)=\\frac1{2N}\\sum_{i=1}^N\n\\frac{\\mathbb E\\operatorname{tr}(e^{(1-u)A_{N,M,\\theta}}q_i e^{uA_{N,M,\\theta}}q_i)}\n{\\mathbb E\\operatorname{tr}e^{A_{N,M,\\theta}}}.\n\\]\nThere exists a family \\((\\mu_\\theta,\\rho_\\theta)_{\\theta\\in[0,1]}\\) satisfying the Dyson equation and\n\\[\n\\Sigma_{\\rho_\\theta}(u)=(1-\\theta)\\Sigma^{(M)}(u)+\\theta\\beta^2G_{\\mu_\\theta}(u)^3\n\\qquad(0\\le u\\le1),\n\\]\nsuch that, for every finite \\(1\\le r<\\infty\\),\n\\[\n\\lim_{\\substack{N\\to\\infty\\\\N\\ {\\rm even}}}\\sup_{\\theta\\in[0,1]}\n\\|G_{N,M,\\theta}-G_{\\mu_\\theta}\\|_{L^r(0,1)}=0.\n\\]"
  },
  "protocol": "single"
}