{
  "statement": "Prove 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\\]",
  "starting_statement": null,
  "target_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\\]",
  "case_id": "SP-1-H0-to-H4",
  "given_context": {
    "premise": null
  },
  "protocol": "single"
}