{
  "statement": "H1: For \\(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\\), there is exactly one such pair satisfying\n\\[\n\\Sigma_\\rho(u)=\\beta^2G_\\mu(u)^3\\qquad(0\\le u\\le1).\n\\]\n\nProve H2: 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 a set \\(I\\subset\\{1,\\ldots,N\\}\\), let \\(A^{\\rm del}\\) be obtained from \\(A_{N,M,\\theta}\\) by omitting the random quartic terms with \\(|S\\cap I|\\ge2\\). For \\(\\sigma\\in\\{\\pm1\\}^I\\), define\n\\[\nA_\\sigma^{\\rm del}=\\alpha_\\sigma(A^{\\rm del}),\\qquad\n\\alpha_\\sigma=\\mathop{\\prod}_{i\\in I:\\sigma_i=-1}\\operatorname{Ad}(q_i),\\qquad\n\\operatorname{Ad}(q_i)(B)=q_iBq_i,\n\\]\nwhere the product is taken in increasing label order. If \\(\\sigma^{(i)}\\) is obtained by reversing its \\(i\\)-th sign, set\n\\[\nX_{i,\\sigma}=\\tfrac12(A_\\sigma^{\\rm del}-A_{\\sigma^{(i)}}^{\\rm del}),\\qquad\nY_{i,\\sigma}(t)=e^{-\\mathrm itA_\\sigma^{\\rm del}}X_{i,\\sigma}e^{\\mathrm itA_\\sigma^{\\rm del}},\n\\]\n\\[\n\\Phi_\\sigma(F)=\\frac{\\mathbb E\\operatorname{tr}(e^{A_\\sigma^{\\rm del}}F)}{\\mathbb E\\operatorname{tr}e^{A_\\sigma^{\\rm del}}}.\n\\]\nLet \\(\\mathfrak H_{m,T}\\) consist of products \\(L=\\prod_{\\ell=1}^s e^{\\mathrm it_\\ell A_{\\tau_\\ell}^{\\rm del}}\\), where \\(0\\le s\\le m\\), \\(\\tau_\\ell\\in\\{\\pm1\\}^I\\), and \\(\\sum_\\ell|t_\\ell|\\le T\\).\nFor every fixed integer \\(k\\ge2\\), integer \\(m\\ge0\\), and finite \\(T,R\\ge0\\),\n\\[\n\\lim_{\\substack{N\\to\\infty\\\\N\\ {\\rm even}}}\n\\sup_{\\substack{|I|=k,\\ i,j\\in I,\\ i\\ne j\\\\\n\\theta\\in[0,1],\\ \\sigma\\in\\{\\pm1\\}^I,\\ L\\in\\mathfrak H_{m,T}\\\\|s|,|t|\\le R}}\n\\Phi_\\sigma\\left(L C_{ij,\\sigma}(s,t)C_{ij,\\sigma}(s,t)^*L^*\\right)=0,\n\\]\nwhere \\(C_{ij,\\sigma}(s,t)=[Y_{i,\\sigma}(s),Y_{j,\\sigma}(t)]\\).",
  "starting_statement": "For \\(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\\), there is exactly one such pair satisfying\n\\[\n\\Sigma_\\rho(u)=\\beta^2G_\\mu(u)^3\\qquad(0\\le u\\le1).\n\\]",
  "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 a set \\(I\\subset\\{1,\\ldots,N\\}\\), let \\(A^{\\rm del}\\) be obtained from \\(A_{N,M,\\theta}\\) by omitting the random quartic terms with \\(|S\\cap I|\\ge2\\). For \\(\\sigma\\in\\{\\pm1\\}^I\\), define\n\\[\nA_\\sigma^{\\rm del}=\\alpha_\\sigma(A^{\\rm del}),\\qquad\n\\alpha_\\sigma=\\mathop{\\prod}_{i\\in I:\\sigma_i=-1}\\operatorname{Ad}(q_i),\\qquad\n\\operatorname{Ad}(q_i)(B)=q_iBq_i,\n\\]\nwhere the product is taken in increasing label order. If \\(\\sigma^{(i)}\\) is obtained by reversing its \\(i\\)-th sign, set\n\\[\nX_{i,\\sigma}=\\tfrac12(A_\\sigma^{\\rm del}-A_{\\sigma^{(i)}}^{\\rm del}),\\qquad\nY_{i,\\sigma}(t)=e^{-\\mathrm itA_\\sigma^{\\rm del}}X_{i,\\sigma}e^{\\mathrm itA_\\sigma^{\\rm del}},\n\\]\n\\[\n\\Phi_\\sigma(F)=\\frac{\\mathbb E\\operatorname{tr}(e^{A_\\sigma^{\\rm del}}F)}{\\mathbb E\\operatorname{tr}e^{A_\\sigma^{\\rm del}}}.\n\\]\nLet \\(\\mathfrak H_{m,T}\\) consist of products \\(L=\\prod_{\\ell=1}^s e^{\\mathrm it_\\ell A_{\\tau_\\ell}^{\\rm del}}\\), where \\(0\\le s\\le m\\), \\(\\tau_\\ell\\in\\{\\pm1\\}^I\\), and \\(\\sum_\\ell|t_\\ell|\\le T\\).\nFor every fixed integer \\(k\\ge2\\), integer \\(m\\ge0\\), and finite \\(T,R\\ge0\\),\n\\[\n\\lim_{\\substack{N\\to\\infty\\\\N\\ {\\rm even}}}\n\\sup_{\\substack{|I|=k,\\ i,j\\in I,\\ i\\ne j\\\\\n\\theta\\in[0,1],\\ \\sigma\\in\\{\\pm1\\}^I,\\ L\\in\\mathfrak H_{m,T}\\\\|s|,|t|\\le R}}\n\\Phi_\\sigma\\left(L C_{ij,\\sigma}(s,t)C_{ij,\\sigma}(s,t)^*L^*\\right)=0,\n\\]\nwhere \\(C_{ij,\\sigma}(s,t)=[Y_{i,\\sigma}(s),Y_{j,\\sigma}(t)]\\).",
  "case_id": "SP-1-H1-to-H2",
  "given_context": {
    "premise": "For \\(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\\), there is exactly one such pair satisfying\n\\[\n\\Sigma_\\rho(u)=\\beta^2G_\\mu(u)^3\\qquad(0\\le u\\le1).\n\\]"
  },
  "protocol": "single"
}