{
  "statement": "Prove 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": null,
  "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-H0-to-H6",
  "given_context": {
    "premise": null
  },
  "protocol": "single"
}