{
  "statement": "H3: 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 probability measure \\(\\mu\\) on \\([0,\\infty)\\), write\n\\[\nG_\\mu(u)=\\int_0^\\infty\\frac{\\cosh(x(1/2-u))}{2\\cosh(x/2)}\\,\\mu(dx),\\qquad0\\le u\\le1.\n\\]\n\nLet \\(B_{N,M,\\theta}\\) be obtained from \\(A_{N,M,\\theta}\\) by removing every random quartic term incident to site \\(N\\), while retaining all bath terms.\nLet \\(r\\ge1\\), let \\(\\mathfrak p\\) be a pairing of \\(\\{1,\\ldots,2r\\}\\), and assign a distinct bulk label \\(i\\in\\{1,\\ldots,N-1\\}\\) to each pair. Write \\(j_a\\) for the label of the pair containing position \\(a\\).\nFor \\(0<t_1<\\cdots<t_{2r}<1\\), define\n\\[\nW_N(\\mathbf t,\\mathfrak p,\\mathbf i)=\n\\frac{\\mathbb E\\operatorname{tr}\\left(\ne^{t_1B}q_{j_1}e^{(t_2-t_1)B}q_{j_2}\\cdots\ne^{(t_{2r}-t_{2r-1})B}q_{j_{2r}}e^{(1-t_{2r})B}\\right)}\n{\\mathbb E\\operatorname{tr}e^B},\\qquad B=B_{N,M,\\theta}.\n\\]\nPut \\(\\varepsilon(\\mathfrak p)=(-1)^{\\operatorname{cr}(\\mathfrak p)}\\), where \\(\\operatorname{cr}\\) counts pairs of arcs with endpoints \\(a<c<b<d\\).\nFor every sequence of even \\(N_j\\to\\infty\\) and \\(\\theta_j\\to\\theta_\\infty\\in[0,1]\\), there exist a subsequence and one Borel probability measure \\(\\Pi\\) on \\(\\mathcal P([0,\\infty))\\), with its weak topology, such that simultaneously for all fixed \\(r\\) and \\(\\mathfrak p\\),\n\\[\nW_{N_j}(\\mathbf t,\\mathfrak p,\\mathbf i)\\longrightarrow\n\\varepsilon(\\mathfrak p)\\int_{\\mathcal P([0,\\infty))}\n\\prod_{\\{a,b\\}\\in\\mathfrak p,\\ a<b}2G_\\mu(t_b-t_a)\\,\\Pi(d\\mu).\n\\]\nConvergence is uniform over the distinct bulk labels and locally uniform in the open simplex \\(0<t_1<\\cdots<t_{2r}<1\\). The same \\(\\Pi\\) applies to every \\(r\\), pairing, and parameter choice.\n\nProve H5: 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, using normalized trace,\n\\[\np_N(\\beta)=\\frac1N\\log\\mathbb E\\operatorname{tr}\n\\exp\\left(\\beta\\sqrt{\\frac3{8N^3}}\\sum_{|S|=4}J_S\\Psi_S\\right).\n\\]\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\nFor every \\(\\beta>0\\), the sequence \\(p_N(\\beta)\\), along even \\(N\\), has a finite limit \\(p(\\beta)\\). For every pair \\((\\mu,\\rho)\\) satisfying the Dyson equation and \\(\\Sigma_\\rho=\\beta^2G_\\mu^3\\), that limit is\n\\[\np(\\beta)=\\sum_{n=0}^\\infty\\log\\left(1+\\int_0^\\infty\\frac{\\rho(dx)}{((2n+1)\\pi)^2+x^2}\\right)\n-\\frac{3\\beta^2}{8}\\int_0^1G_\\mu(u)^4\\,du.\n\\]",
  "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 a probability measure \\(\\mu\\) on \\([0,\\infty)\\), write\n\\[\nG_\\mu(u)=\\int_0^\\infty\\frac{\\cosh(x(1/2-u))}{2\\cosh(x/2)}\\,\\mu(dx),\\qquad0\\le u\\le1.\n\\]\n\nLet \\(B_{N,M,\\theta}\\) be obtained from \\(A_{N,M,\\theta}\\) by removing every random quartic term incident to site \\(N\\), while retaining all bath terms.\nLet \\(r\\ge1\\), let \\(\\mathfrak p\\) be a pairing of \\(\\{1,\\ldots,2r\\}\\), and assign a distinct bulk label \\(i\\in\\{1,\\ldots,N-1\\}\\) to each pair. Write \\(j_a\\) for the label of the pair containing position \\(a\\).\nFor \\(0<t_1<\\cdots<t_{2r}<1\\), define\n\\[\nW_N(\\mathbf t,\\mathfrak p,\\mathbf i)=\n\\frac{\\mathbb E\\operatorname{tr}\\left(\ne^{t_1B}q_{j_1}e^{(t_2-t_1)B}q_{j_2}\\cdots\ne^{(t_{2r}-t_{2r-1})B}q_{j_{2r}}e^{(1-t_{2r})B}\\right)}\n{\\mathbb E\\operatorname{tr}e^B},\\qquad B=B_{N,M,\\theta}.\n\\]\nPut \\(\\varepsilon(\\mathfrak p)=(-1)^{\\operatorname{cr}(\\mathfrak p)}\\), where \\(\\operatorname{cr}\\) counts pairs of arcs with endpoints \\(a<c<b<d\\).\nFor every sequence of even \\(N_j\\to\\infty\\) and \\(\\theta_j\\to\\theta_\\infty\\in[0,1]\\), there exist a subsequence and one Borel probability measure \\(\\Pi\\) on \\(\\mathcal P([0,\\infty))\\), with its weak topology, such that simultaneously for all fixed \\(r\\) and \\(\\mathfrak p\\),\n\\[\nW_{N_j}(\\mathbf t,\\mathfrak p,\\mathbf i)\\longrightarrow\n\\varepsilon(\\mathfrak p)\\int_{\\mathcal P([0,\\infty))}\n\\prod_{\\{a,b\\}\\in\\mathfrak p,\\ a<b}2G_\\mu(t_b-t_a)\\,\\Pi(d\\mu).\n\\]\nConvergence is uniform over the distinct bulk labels and locally uniform in the open simplex \\(0<t_1<\\cdots<t_{2r}<1\\). The same \\(\\Pi\\) applies to every \\(r\\), pairing, and parameter choice.",
  "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, using normalized trace,\n\\[\np_N(\\beta)=\\frac1N\\log\\mathbb E\\operatorname{tr}\n\\exp\\left(\\beta\\sqrt{\\frac3{8N^3}}\\sum_{|S|=4}J_S\\Psi_S\\right).\n\\]\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\nFor every \\(\\beta>0\\), the sequence \\(p_N(\\beta)\\), along even \\(N\\), has a finite limit \\(p(\\beta)\\). For every pair \\((\\mu,\\rho)\\) satisfying the Dyson equation and \\(\\Sigma_\\rho=\\beta^2G_\\mu^3\\), that limit is\n\\[\np(\\beta)=\\sum_{n=0}^\\infty\\log\\left(1+\\int_0^\\infty\\frac{\\rho(dx)}{((2n+1)\\pi)^2+x^2}\\right)\n-\\frac{3\\beta^2}{8}\\int_0^1G_\\mu(u)^4\\,du.\n\\]",
  "case_id": "SP-1-H3-to-H5",
  "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 a probability measure \\(\\mu\\) on \\([0,\\infty)\\), write\n\\[\nG_\\mu(u)=\\int_0^\\infty\\frac{\\cosh(x(1/2-u))}{2\\cosh(x/2)}\\,\\mu(dx),\\qquad0\\le u\\le1.\n\\]\n\nLet \\(B_{N,M,\\theta}\\) be obtained from \\(A_{N,M,\\theta}\\) by removing every random quartic term incident to site \\(N\\), while retaining all bath terms.\nLet \\(r\\ge1\\), let \\(\\mathfrak p\\) be a pairing of \\(\\{1,\\ldots,2r\\}\\), and assign a distinct bulk label \\(i\\in\\{1,\\ldots,N-1\\}\\) to each pair. Write \\(j_a\\) for the label of the pair containing position \\(a\\).\nFor \\(0<t_1<\\cdots<t_{2r}<1\\), define\n\\[\nW_N(\\mathbf t,\\mathfrak p,\\mathbf i)=\n\\frac{\\mathbb E\\operatorname{tr}\\left(\ne^{t_1B}q_{j_1}e^{(t_2-t_1)B}q_{j_2}\\cdots\ne^{(t_{2r}-t_{2r-1})B}q_{j_{2r}}e^{(1-t_{2r})B}\\right)}\n{\\mathbb E\\operatorname{tr}e^B},\\qquad B=B_{N,M,\\theta}.\n\\]\nPut \\(\\varepsilon(\\mathfrak p)=(-1)^{\\operatorname{cr}(\\mathfrak p)}\\), where \\(\\operatorname{cr}\\) counts pairs of arcs with endpoints \\(a<c<b<d\\).\nFor every sequence of even \\(N_j\\to\\infty\\) and \\(\\theta_j\\to\\theta_\\infty\\in[0,1]\\), there exist a subsequence and one Borel probability measure \\(\\Pi\\) on \\(\\mathcal P([0,\\infty))\\), with its weak topology, such that simultaneously for all fixed \\(r\\) and \\(\\mathfrak p\\),\n\\[\nW_{N_j}(\\mathbf t,\\mathfrak p,\\mathbf i)\\longrightarrow\n\\varepsilon(\\mathfrak p)\\int_{\\mathcal P([0,\\infty))}\n\\prod_{\\{a,b\\}\\in\\mathfrak p,\\ a<b}2G_\\mu(t_b-t_a)\\,\\Pi(d\\mu).\n\\]\nConvergence is uniform over the distinct bulk labels and locally uniform in the open simplex \\(0<t_1<\\cdots<t_{2r}<1\\). The same \\(\\Pi\\) applies to every \\(r\\), pairing, and parameter choice."
  },
  "protocol": "single"
}