{
  "statement": "Prove 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\\]",
  "starting_statement": null,
  "target_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\\]",
  "case_id": "SP-1-H0-to-H1",
  "given_context": {
    "premise": null
  },
  "protocol": "single"
}