{
  "EP-1-H0-to-H1": {
    "id": "EP-1-H0-to-H1",
    "problem_id": "EP-1",
    "problem_name": "Berenstein rigidity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 1,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H1: Let \\(m\\ge2\\) be an integer and let \\(\\rho>1\\). Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution, when it exists, of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor every integer \\(m\\ge2\\) and every \\(\\rho>1\\), the coefficient series define continuous functions on their respective closed domains. The spaces \\(W_{m,\\rho}\\), \\(Y_{m,\\rho}\\), \\(P_{m,\\rho}\\), and \\(B_{m,\\rho}\\) are Banach algebras. Their multiplication and supremum-norm embedding constants are at most one.\nFor every \\(g\\in Y_{m,\\rho}\\), the weak Dirichlet solution exists and belongs to\n\\(Y_{m,\\rho}\\cap W^{2,q}(D)\\cap W_0^{1,q}(D)\\) for every finite \\(q>1\\). The operators satisfy\n\\[\n\\|K_D\\|_{Y_{m,\\rho}\\to Y_{m,\\rho}}=\\frac14,\n\\qquad\n\\|N\\|_{Y_{m,\\rho}\\to B_{m,\\rho}}=\\frac12.\n\\]\nWith the operators extended complex linearly,\n\\[\nN\\Phi_{\\ell,0}=\\frac{e^{im\\ell\\theta}}{2(m|\\ell|+1)},\n\\qquad N\\Phi_{\\ell,s}=0\\quad(s\\ge1).\n\\]",
    "starting_statement": null,
    "target_statement": "Let \\(m\\ge2\\) be an integer and let \\(\\rho>1\\). Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution, when it exists, of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor every integer \\(m\\ge2\\) and every \\(\\rho>1\\), the coefficient series define continuous functions on their respective closed domains. The spaces \\(W_{m,\\rho}\\), \\(Y_{m,\\rho}\\), \\(P_{m,\\rho}\\), and \\(B_{m,\\rho}\\) are Banach algebras. Their multiplication and supremum-norm embedding constants are at most one.\nFor every \\(g\\in Y_{m,\\rho}\\), the weak Dirichlet solution exists and belongs to\n\\(Y_{m,\\rho}\\cap W^{2,q}(D)\\cap W_0^{1,q}(D)\\) for every finite \\(q>1\\). The operators satisfy\n\\[\n\\|K_D\\|_{Y_{m,\\rho}\\to Y_{m,\\rho}}=\\frac14,\n\\qquad\n\\|N\\|_{Y_{m,\\rho}\\to B_{m,\\rho}}=\\frac12.\n\\]\nWith the operators extended complex linearly,\n\\[\nN\\Phi_{\\ell,0}=\\frac{e^{im\\ell\\theta}}{2(m|\\ell|+1)},\n\\qquad N\\Phi_{\\ell,s}=0\\quad(s\\ge1).\n\\]",
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  "EP-1-H0-to-H2": {
    "id": "EP-1-H0-to-H2",
    "problem_id": "EP-1",
    "problem_name": "Berenstein rigidity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 2,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H2: Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). Let \\(\\phi\\) be holomorphic on a neighborhood of \\(\\overline D\\), with\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D).\n\\]\nSuppose \\(k>0\\) and a real-valued function \\(U\\in C^2(\\overline D)\\) satisfy\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.\n\nThen \\(\\phi\\) is injective on a neighborhood of \\(\\overline D\\). The set \\(\\Omega=\\phi(D)\\) is a bounded simply connected domain with real-analytic Jordan boundary, and\n\\[\nu=U\\circ\\phi^{-1}\n\\]\nis a nonzero real-valued function in \\(C^2(\\overline\\Omega)\\) satisfying\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
    "starting_statement": null,
    "target_statement": "Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). Let \\(\\phi\\) be holomorphic on a neighborhood of \\(\\overline D\\), with\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D).\n\\]\nSuppose \\(k>0\\) and a real-valued function \\(U\\in C^2(\\overline D)\\) satisfy\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.\n\nThen \\(\\phi\\) is injective on a neighborhood of \\(\\overline D\\). The set \\(\\Omega=\\phi(D)\\) is a bounded simply connected domain with real-analytic Jordan boundary, and\n\\[\nu=U\\circ\\phi^{-1}\n\\]\nis a nonzero real-valued function in \\(C^2(\\overline\\Omega)\\) satisfying\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
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  "EP-1-H0-to-H3": {
    "id": "EP-1-H0-to-H3",
    "problem_id": "EP-1",
    "problem_name": "Berenstein rigidity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H3: For each integer \\(m\\ge2\\) and each \\(\\rho>1\\), define the following spaces and operators. Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor \\(g\\in Y_{m,\\rho}\\), \\(k\\in\\mathbb R\\), and \\(a\\in P_{m,\\rho}\\) with \\(a_0=1\\), define\n\\[\nF_{m,\\rho}(g,k,a)=\\bigl(g+k^2|a|^2K_Dg,\\ (Ng)^2-|a|^2|_{\\partial D}\\bigr).\n\\]\nIts affine domain is\n\\[\nX_{m,\\rho}=\\{(g,k,a)\\in Y_{m,\\rho}\\times\\mathbb R\\times P_{m,\\rho}:a_0=1\\},\n\\]\nwith tangent space \\(X_{m,\\rho}^{\\rm tan}=Y_{m,\\rho}\\times\\mathbb R\\times\\{\\dot a\\in P_{m,\\rho}:\\dot a_0=0\\}\\) and norm\n\\[\n\\|(\\dot g,\\dot k,\\dot a)\\|_X=\\|\\dot g\\|_\\rho+|\\dot k|+\\|\\dot a\\|_P.\n\\]\nIts codomain is \\(Z_{m,\\rho}=Y_{m,\\rho}\\times B_{m,\\rho}\\), with norm\n\\(\\|(y,b)\\|_Z=\\|y\\|_\\rho+\\|b\\|_B\\).\n\nThere exist an integer \\(m\\ge2\\), a real number \\(\\rho>1\\), a point \\(x_c=(g_c,k_c,a_c)\\in X_{m,\\rho}\\) with \\(g_c\\) and \\(a_c\\) of finite coefficient support, a bounded injective real-linear operator\n\\[\nA:Z_{m,\\rho}\\longrightarrow X_{m,\\rho}^{\\rm tan},\n\\]\nand real numbers \\(r>0\\) and \\(0\\le q<1\\) with the following properties.\nWrite\n\\[\nB_r(x_c)=\\{x\\in X_{m,\\rho}:\\|x-x_c\\|_X\\le r\\}.\n\\]\nThe map \\(F_{m,\\rho}\\) is well-defined and continuously Fréchet differentiable on an open neighborhood of \\(B_r(x_c)\\), and\n\\[\n\\|A F_{m,\\rho}(x_c)\\|_X<(1-q)r,\n\\qquad\n\\sup_{x\\in B_r(x_c)}\n\\|I-A\\,DF_{m,\\rho}(x)\\|_{X_{m,\\rho}^{\\rm tan}\\to X_{m,\\rho}^{\\rm tan}}\\le q,\n\\]\nwhere \\(I\\) is the identity on \\(X_{m,\\rho}^{\\rm tan}\\).\nFor every \\((g,k,a)\\in B_r(x_c)\\),\n\\[\nk>0,\\qquad\n\\operatorname{Re}a(z)>0\\quad(z\\in\\overline D),\\qquad\nNg(e^{i\\theta})>0\\quad(\\theta\\in\\mathbb R),\\qquad a_1\\ne0.\n\\]",
    "starting_statement": null,
    "target_statement": "For each integer \\(m\\ge2\\) and each \\(\\rho>1\\), define the following spaces and operators. Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor \\(g\\in Y_{m,\\rho}\\), \\(k\\in\\mathbb R\\), and \\(a\\in P_{m,\\rho}\\) with \\(a_0=1\\), define\n\\[\nF_{m,\\rho}(g,k,a)=\\bigl(g+k^2|a|^2K_Dg,\\ (Ng)^2-|a|^2|_{\\partial D}\\bigr).\n\\]\nIts affine domain is\n\\[\nX_{m,\\rho}=\\{(g,k,a)\\in Y_{m,\\rho}\\times\\mathbb R\\times P_{m,\\rho}:a_0=1\\},\n\\]\nwith tangent space \\(X_{m,\\rho}^{\\rm tan}=Y_{m,\\rho}\\times\\mathbb R\\times\\{\\dot a\\in P_{m,\\rho}:\\dot a_0=0\\}\\) and norm\n\\[\n\\|(\\dot g,\\dot k,\\dot a)\\|_X=\\|\\dot g\\|_\\rho+|\\dot k|+\\|\\dot a\\|_P.\n\\]\nIts codomain is \\(Z_{m,\\rho}=Y_{m,\\rho}\\times B_{m,\\rho}\\), with norm\n\\(\\|(y,b)\\|_Z=\\|y\\|_\\rho+\\|b\\|_B\\).\n\nThere exist an integer \\(m\\ge2\\), a real number \\(\\rho>1\\), a point \\(x_c=(g_c,k_c,a_c)\\in X_{m,\\rho}\\) with \\(g_c\\) and \\(a_c\\) of finite coefficient support, a bounded injective real-linear operator\n\\[\nA:Z_{m,\\rho}\\longrightarrow X_{m,\\rho}^{\\rm tan},\n\\]\nand real numbers \\(r>0\\) and \\(0\\le q<1\\) with the following properties.\nWrite\n\\[\nB_r(x_c)=\\{x\\in X_{m,\\rho}:\\|x-x_c\\|_X\\le r\\}.\n\\]\nThe map \\(F_{m,\\rho}\\) is well-defined and continuously Fréchet differentiable on an open neighborhood of \\(B_r(x_c)\\), and\n\\[\n\\|A F_{m,\\rho}(x_c)\\|_X<(1-q)r,\n\\qquad\n\\sup_{x\\in B_r(x_c)}\n\\|I-A\\,DF_{m,\\rho}(x)\\|_{X_{m,\\rho}^{\\rm tan}\\to X_{m,\\rho}^{\\rm tan}}\\le q,\n\\]\nwhere \\(I\\) is the identity on \\(X_{m,\\rho}^{\\rm tan}\\).\nFor every \\((g,k,a)\\in B_r(x_c)\\),\n\\[\nk>0,\\qquad\n\\operatorname{Re}a(z)>0\\quad(z\\in\\overline D),\\qquad\nNg(e^{i\\theta})>0\\quad(\\theta\\in\\mathbb R),\\qquad a_1\\ne0.\n\\]",
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  "EP-1-H0-to-H4": {
    "id": "EP-1-H0-to-H4",
    "problem_id": "EP-1",
    "problem_name": "Berenstein rigidity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H4: Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). There exist an integer \\(m\\ge2\\), a real number \\(k>0\\), a holomorphic function \\(\\phi\\) on a neighborhood of \\(\\overline D\\), and a real-valued function \\(U\\in C^2(\\overline D)\\), such that\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D),\n\\]\n\\[\n\\phi(e^{2\\pi i/m}z)=e^{2\\pi i/m}\\phi(z)\\quad(z\\in D),\n\\qquad \\phi\\not\\equiv z,\n\\]\nand\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.",
    "starting_statement": null,
    "target_statement": "Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). There exist an integer \\(m\\ge2\\), a real number \\(k>0\\), a holomorphic function \\(\\phi\\) on a neighborhood of \\(\\overline D\\), and a real-valued function \\(U\\in C^2(\\overline D)\\), such that\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D),\n\\]\n\\[\n\\phi(e^{2\\pi i/m}z)=e^{2\\pi i/m}\\phi(z)\\quad(z\\in D),\n\\qquad \\phi\\not\\equiv z,\n\\]\nand\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.",
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  "EP-1-H0-to-H5": {
    "id": "EP-1-H0-to-H5",
    "problem_id": "EP-1",
    "problem_name": "Berenstein rigidity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H5: There exist a bounded simply connected domain \\(\\Omega\\subset\\mathbb R^2\\) with \\(C^\\infty\\) boundary, a real number \\(k>0\\), and a nonzero real-valued function \\(u\\in C^2(\\overline\\Omega)\\), such that \\(\\Omega\\) is not a disk and\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
    "starting_statement": null,
    "target_statement": "There exist a bounded simply connected domain \\(\\Omega\\subset\\mathbb R^2\\) with \\(C^\\infty\\) boundary, a real number \\(k>0\\), and a nonzero real-valued function \\(u\\in C^2(\\overline\\Omega)\\), such that \\(\\Omega\\) is not a disk and\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
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  "EP-1-H1-to-H2": {
    "id": "EP-1-H1-to-H2",
    "problem_id": "EP-1",
    "problem_name": "Berenstein rigidity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 2,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(m\\ge2\\) be an integer and let \\(\\rho>1\\). Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution, when it exists, of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor every integer \\(m\\ge2\\) and every \\(\\rho>1\\), the coefficient series define continuous functions on their respective closed domains. The spaces \\(W_{m,\\rho}\\), \\(Y_{m,\\rho}\\), \\(P_{m,\\rho}\\), and \\(B_{m,\\rho}\\) are Banach algebras. Their multiplication and supremum-norm embedding constants are at most one.\nFor every \\(g\\in Y_{m,\\rho}\\), the weak Dirichlet solution exists and belongs to\n\\(Y_{m,\\rho}\\cap W^{2,q}(D)\\cap W_0^{1,q}(D)\\) for every finite \\(q>1\\). The operators satisfy\n\\[\n\\|K_D\\|_{Y_{m,\\rho}\\to Y_{m,\\rho}}=\\frac14,\n\\qquad\n\\|N\\|_{Y_{m,\\rho}\\to B_{m,\\rho}}=\\frac12.\n\\]\nWith the operators extended complex linearly,\n\\[\nN\\Phi_{\\ell,0}=\\frac{e^{im\\ell\\theta}}{2(m|\\ell|+1)},\n\\qquad N\\Phi_{\\ell,s}=0\\quad(s\\ge1).\n\\]\n\nProve H2: Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). Let \\(\\phi\\) be holomorphic on a neighborhood of \\(\\overline D\\), with\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D).\n\\]\nSuppose \\(k>0\\) and a real-valued function \\(U\\in C^2(\\overline D)\\) satisfy\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.\n\nThen \\(\\phi\\) is injective on a neighborhood of \\(\\overline D\\). The set \\(\\Omega=\\phi(D)\\) is a bounded simply connected domain with real-analytic Jordan boundary, and\n\\[\nu=U\\circ\\phi^{-1}\n\\]\nis a nonzero real-valued function in \\(C^2(\\overline\\Omega)\\) satisfying\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
    "starting_statement": "Let \\(m\\ge2\\) be an integer and let \\(\\rho>1\\). Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution, when it exists, of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor every integer \\(m\\ge2\\) and every \\(\\rho>1\\), the coefficient series define continuous functions on their respective closed domains. The spaces \\(W_{m,\\rho}\\), \\(Y_{m,\\rho}\\), \\(P_{m,\\rho}\\), and \\(B_{m,\\rho}\\) are Banach algebras. Their multiplication and supremum-norm embedding constants are at most one.\nFor every \\(g\\in Y_{m,\\rho}\\), the weak Dirichlet solution exists and belongs to\n\\(Y_{m,\\rho}\\cap W^{2,q}(D)\\cap W_0^{1,q}(D)\\) for every finite \\(q>1\\). The operators satisfy\n\\[\n\\|K_D\\|_{Y_{m,\\rho}\\to Y_{m,\\rho}}=\\frac14,\n\\qquad\n\\|N\\|_{Y_{m,\\rho}\\to B_{m,\\rho}}=\\frac12.\n\\]\nWith the operators extended complex linearly,\n\\[\nN\\Phi_{\\ell,0}=\\frac{e^{im\\ell\\theta}}{2(m|\\ell|+1)},\n\\qquad N\\Phi_{\\ell,s}=0\\quad(s\\ge1).\n\\]",
    "target_statement": "Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). Let \\(\\phi\\) be holomorphic on a neighborhood of \\(\\overline D\\), with\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D).\n\\]\nSuppose \\(k>0\\) and a real-valued function \\(U\\in C^2(\\overline D)\\) satisfy\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.\n\nThen \\(\\phi\\) is injective on a neighborhood of \\(\\overline D\\). The set \\(\\Omega=\\phi(D)\\) is a bounded simply connected domain with real-analytic Jordan boundary, and\n\\[\nu=U\\circ\\phi^{-1}\n\\]\nis a nonzero real-valued function in \\(C^2(\\overline\\Omega)\\) satisfying\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
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  "EP-1-H1-to-H3": {
    "id": "EP-1-H1-to-H3",
    "problem_id": "EP-1",
    "problem_name": "Berenstein rigidity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(m\\ge2\\) be an integer and let \\(\\rho>1\\). Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution, when it exists, of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor every integer \\(m\\ge2\\) and every \\(\\rho>1\\), the coefficient series define continuous functions on their respective closed domains. The spaces \\(W_{m,\\rho}\\), \\(Y_{m,\\rho}\\), \\(P_{m,\\rho}\\), and \\(B_{m,\\rho}\\) are Banach algebras. Their multiplication and supremum-norm embedding constants are at most one.\nFor every \\(g\\in Y_{m,\\rho}\\), the weak Dirichlet solution exists and belongs to\n\\(Y_{m,\\rho}\\cap W^{2,q}(D)\\cap W_0^{1,q}(D)\\) for every finite \\(q>1\\). The operators satisfy\n\\[\n\\|K_D\\|_{Y_{m,\\rho}\\to Y_{m,\\rho}}=\\frac14,\n\\qquad\n\\|N\\|_{Y_{m,\\rho}\\to B_{m,\\rho}}=\\frac12.\n\\]\nWith the operators extended complex linearly,\n\\[\nN\\Phi_{\\ell,0}=\\frac{e^{im\\ell\\theta}}{2(m|\\ell|+1)},\n\\qquad N\\Phi_{\\ell,s}=0\\quad(s\\ge1).\n\\]\n\nProve H3: For each integer \\(m\\ge2\\) and each \\(\\rho>1\\), define the following spaces and operators. Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor \\(g\\in Y_{m,\\rho}\\), \\(k\\in\\mathbb R\\), and \\(a\\in P_{m,\\rho}\\) with \\(a_0=1\\), define\n\\[\nF_{m,\\rho}(g,k,a)=\\bigl(g+k^2|a|^2K_Dg,\\ (Ng)^2-|a|^2|_{\\partial D}\\bigr).\n\\]\nIts affine domain is\n\\[\nX_{m,\\rho}=\\{(g,k,a)\\in Y_{m,\\rho}\\times\\mathbb R\\times P_{m,\\rho}:a_0=1\\},\n\\]\nwith tangent space \\(X_{m,\\rho}^{\\rm tan}=Y_{m,\\rho}\\times\\mathbb R\\times\\{\\dot a\\in P_{m,\\rho}:\\dot a_0=0\\}\\) and norm\n\\[\n\\|(\\dot g,\\dot k,\\dot a)\\|_X=\\|\\dot g\\|_\\rho+|\\dot k|+\\|\\dot a\\|_P.\n\\]\nIts codomain is \\(Z_{m,\\rho}=Y_{m,\\rho}\\times B_{m,\\rho}\\), with norm\n\\(\\|(y,b)\\|_Z=\\|y\\|_\\rho+\\|b\\|_B\\).\n\nThere exist an integer \\(m\\ge2\\), a real number \\(\\rho>1\\), a point \\(x_c=(g_c,k_c,a_c)\\in X_{m,\\rho}\\) with \\(g_c\\) and \\(a_c\\) of finite coefficient support, a bounded injective real-linear operator\n\\[\nA:Z_{m,\\rho}\\longrightarrow X_{m,\\rho}^{\\rm tan},\n\\]\nand real numbers \\(r>0\\) and \\(0\\le q<1\\) with the following properties.\nWrite\n\\[\nB_r(x_c)=\\{x\\in X_{m,\\rho}:\\|x-x_c\\|_X\\le r\\}.\n\\]\nThe map \\(F_{m,\\rho}\\) is well-defined and continuously Fréchet differentiable on an open neighborhood of \\(B_r(x_c)\\), and\n\\[\n\\|A F_{m,\\rho}(x_c)\\|_X<(1-q)r,\n\\qquad\n\\sup_{x\\in B_r(x_c)}\n\\|I-A\\,DF_{m,\\rho}(x)\\|_{X_{m,\\rho}^{\\rm tan}\\to X_{m,\\rho}^{\\rm tan}}\\le q,\n\\]\nwhere \\(I\\) is the identity on \\(X_{m,\\rho}^{\\rm tan}\\).\nFor every \\((g,k,a)\\in B_r(x_c)\\),\n\\[\nk>0,\\qquad\n\\operatorname{Re}a(z)>0\\quad(z\\in\\overline D),\\qquad\nNg(e^{i\\theta})>0\\quad(\\theta\\in\\mathbb R),\\qquad a_1\\ne0.\n\\]",
    "starting_statement": "Let \\(m\\ge2\\) be an integer and let \\(\\rho>1\\). Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution, when it exists, of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor every integer \\(m\\ge2\\) and every \\(\\rho>1\\), the coefficient series define continuous functions on their respective closed domains. The spaces \\(W_{m,\\rho}\\), \\(Y_{m,\\rho}\\), \\(P_{m,\\rho}\\), and \\(B_{m,\\rho}\\) are Banach algebras. Their multiplication and supremum-norm embedding constants are at most one.\nFor every \\(g\\in Y_{m,\\rho}\\), the weak Dirichlet solution exists and belongs to\n\\(Y_{m,\\rho}\\cap W^{2,q}(D)\\cap W_0^{1,q}(D)\\) for every finite \\(q>1\\). The operators satisfy\n\\[\n\\|K_D\\|_{Y_{m,\\rho}\\to Y_{m,\\rho}}=\\frac14,\n\\qquad\n\\|N\\|_{Y_{m,\\rho}\\to B_{m,\\rho}}=\\frac12.\n\\]\nWith the operators extended complex linearly,\n\\[\nN\\Phi_{\\ell,0}=\\frac{e^{im\\ell\\theta}}{2(m|\\ell|+1)},\n\\qquad N\\Phi_{\\ell,s}=0\\quad(s\\ge1).\n\\]",
    "target_statement": "For each integer \\(m\\ge2\\) and each \\(\\rho>1\\), define the following spaces and operators. Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor \\(g\\in Y_{m,\\rho}\\), \\(k\\in\\mathbb R\\), and \\(a\\in P_{m,\\rho}\\) with \\(a_0=1\\), define\n\\[\nF_{m,\\rho}(g,k,a)=\\bigl(g+k^2|a|^2K_Dg,\\ (Ng)^2-|a|^2|_{\\partial D}\\bigr).\n\\]\nIts affine domain is\n\\[\nX_{m,\\rho}=\\{(g,k,a)\\in Y_{m,\\rho}\\times\\mathbb R\\times P_{m,\\rho}:a_0=1\\},\n\\]\nwith tangent space \\(X_{m,\\rho}^{\\rm tan}=Y_{m,\\rho}\\times\\mathbb R\\times\\{\\dot a\\in P_{m,\\rho}:\\dot a_0=0\\}\\) and norm\n\\[\n\\|(\\dot g,\\dot k,\\dot a)\\|_X=\\|\\dot g\\|_\\rho+|\\dot k|+\\|\\dot a\\|_P.\n\\]\nIts codomain is \\(Z_{m,\\rho}=Y_{m,\\rho}\\times B_{m,\\rho}\\), with norm\n\\(\\|(y,b)\\|_Z=\\|y\\|_\\rho+\\|b\\|_B\\).\n\nThere exist an integer \\(m\\ge2\\), a real number \\(\\rho>1\\), a point \\(x_c=(g_c,k_c,a_c)\\in X_{m,\\rho}\\) with \\(g_c\\) and \\(a_c\\) of finite coefficient support, a bounded injective real-linear operator\n\\[\nA:Z_{m,\\rho}\\longrightarrow X_{m,\\rho}^{\\rm tan},\n\\]\nand real numbers \\(r>0\\) and \\(0\\le q<1\\) with the following properties.\nWrite\n\\[\nB_r(x_c)=\\{x\\in X_{m,\\rho}:\\|x-x_c\\|_X\\le r\\}.\n\\]\nThe map \\(F_{m,\\rho}\\) is well-defined and continuously Fréchet differentiable on an open neighborhood of \\(B_r(x_c)\\), and\n\\[\n\\|A F_{m,\\rho}(x_c)\\|_X<(1-q)r,\n\\qquad\n\\sup_{x\\in B_r(x_c)}\n\\|I-A\\,DF_{m,\\rho}(x)\\|_{X_{m,\\rho}^{\\rm tan}\\to X_{m,\\rho}^{\\rm tan}}\\le q,\n\\]\nwhere \\(I\\) is the identity on \\(X_{m,\\rho}^{\\rm tan}\\).\nFor every \\((g,k,a)\\in B_r(x_c)\\),\n\\[\nk>0,\\qquad\n\\operatorname{Re}a(z)>0\\quad(z\\in\\overline D),\\qquad\nNg(e^{i\\theta})>0\\quad(\\theta\\in\\mathbb R),\\qquad a_1\\ne0.\n\\]",
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  "EP-1-H1-to-H4": {
    "id": "EP-1-H1-to-H4",
    "problem_id": "EP-1",
    "problem_name": "Berenstein rigidity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(m\\ge2\\) be an integer and let \\(\\rho>1\\). Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution, when it exists, of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor every integer \\(m\\ge2\\) and every \\(\\rho>1\\), the coefficient series define continuous functions on their respective closed domains. The spaces \\(W_{m,\\rho}\\), \\(Y_{m,\\rho}\\), \\(P_{m,\\rho}\\), and \\(B_{m,\\rho}\\) are Banach algebras. Their multiplication and supremum-norm embedding constants are at most one.\nFor every \\(g\\in Y_{m,\\rho}\\), the weak Dirichlet solution exists and belongs to\n\\(Y_{m,\\rho}\\cap W^{2,q}(D)\\cap W_0^{1,q}(D)\\) for every finite \\(q>1\\). The operators satisfy\n\\[\n\\|K_D\\|_{Y_{m,\\rho}\\to Y_{m,\\rho}}=\\frac14,\n\\qquad\n\\|N\\|_{Y_{m,\\rho}\\to B_{m,\\rho}}=\\frac12.\n\\]\nWith the operators extended complex linearly,\n\\[\nN\\Phi_{\\ell,0}=\\frac{e^{im\\ell\\theta}}{2(m|\\ell|+1)},\n\\qquad N\\Phi_{\\ell,s}=0\\quad(s\\ge1).\n\\]\n\nProve H4: Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). There exist an integer \\(m\\ge2\\), a real number \\(k>0\\), a holomorphic function \\(\\phi\\) on a neighborhood of \\(\\overline D\\), and a real-valued function \\(U\\in C^2(\\overline D)\\), such that\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D),\n\\]\n\\[\n\\phi(e^{2\\pi i/m}z)=e^{2\\pi i/m}\\phi(z)\\quad(z\\in D),\n\\qquad \\phi\\not\\equiv z,\n\\]\nand\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.",
    "starting_statement": "Let \\(m\\ge2\\) be an integer and let \\(\\rho>1\\). Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution, when it exists, of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor every integer \\(m\\ge2\\) and every \\(\\rho>1\\), the coefficient series define continuous functions on their respective closed domains. The spaces \\(W_{m,\\rho}\\), \\(Y_{m,\\rho}\\), \\(P_{m,\\rho}\\), and \\(B_{m,\\rho}\\) are Banach algebras. Their multiplication and supremum-norm embedding constants are at most one.\nFor every \\(g\\in Y_{m,\\rho}\\), the weak Dirichlet solution exists and belongs to\n\\(Y_{m,\\rho}\\cap W^{2,q}(D)\\cap W_0^{1,q}(D)\\) for every finite \\(q>1\\). The operators satisfy\n\\[\n\\|K_D\\|_{Y_{m,\\rho}\\to Y_{m,\\rho}}=\\frac14,\n\\qquad\n\\|N\\|_{Y_{m,\\rho}\\to B_{m,\\rho}}=\\frac12.\n\\]\nWith the operators extended complex linearly,\n\\[\nN\\Phi_{\\ell,0}=\\frac{e^{im\\ell\\theta}}{2(m|\\ell|+1)},\n\\qquad N\\Phi_{\\ell,s}=0\\quad(s\\ge1).\n\\]",
    "target_statement": "Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). There exist an integer \\(m\\ge2\\), a real number \\(k>0\\), a holomorphic function \\(\\phi\\) on a neighborhood of \\(\\overline D\\), and a real-valued function \\(U\\in C^2(\\overline D)\\), such that\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D),\n\\]\n\\[\n\\phi(e^{2\\pi i/m}z)=e^{2\\pi i/m}\\phi(z)\\quad(z\\in D),\n\\qquad \\phi\\not\\equiv z,\n\\]\nand\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.",
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  "EP-1-H1-to-H5": {
    "id": "EP-1-H1-to-H5",
    "problem_id": "EP-1",
    "problem_name": "Berenstein rigidity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(m\\ge2\\) be an integer and let \\(\\rho>1\\). Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution, when it exists, of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor every integer \\(m\\ge2\\) and every \\(\\rho>1\\), the coefficient series define continuous functions on their respective closed domains. The spaces \\(W_{m,\\rho}\\), \\(Y_{m,\\rho}\\), \\(P_{m,\\rho}\\), and \\(B_{m,\\rho}\\) are Banach algebras. Their multiplication and supremum-norm embedding constants are at most one.\nFor every \\(g\\in Y_{m,\\rho}\\), the weak Dirichlet solution exists and belongs to\n\\(Y_{m,\\rho}\\cap W^{2,q}(D)\\cap W_0^{1,q}(D)\\) for every finite \\(q>1\\). The operators satisfy\n\\[\n\\|K_D\\|_{Y_{m,\\rho}\\to Y_{m,\\rho}}=\\frac14,\n\\qquad\n\\|N\\|_{Y_{m,\\rho}\\to B_{m,\\rho}}=\\frac12.\n\\]\nWith the operators extended complex linearly,\n\\[\nN\\Phi_{\\ell,0}=\\frac{e^{im\\ell\\theta}}{2(m|\\ell|+1)},\n\\qquad N\\Phi_{\\ell,s}=0\\quad(s\\ge1).\n\\]\n\nProve H5: There exist a bounded simply connected domain \\(\\Omega\\subset\\mathbb R^2\\) with \\(C^\\infty\\) boundary, a real number \\(k>0\\), and a nonzero real-valued function \\(u\\in C^2(\\overline\\Omega)\\), such that \\(\\Omega\\) is not a disk and\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
    "starting_statement": "Let \\(m\\ge2\\) be an integer and let \\(\\rho>1\\). Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution, when it exists, of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor every integer \\(m\\ge2\\) and every \\(\\rho>1\\), the coefficient series define continuous functions on their respective closed domains. The spaces \\(W_{m,\\rho}\\), \\(Y_{m,\\rho}\\), \\(P_{m,\\rho}\\), and \\(B_{m,\\rho}\\) are Banach algebras. Their multiplication and supremum-norm embedding constants are at most one.\nFor every \\(g\\in Y_{m,\\rho}\\), the weak Dirichlet solution exists and belongs to\n\\(Y_{m,\\rho}\\cap W^{2,q}(D)\\cap W_0^{1,q}(D)\\) for every finite \\(q>1\\). The operators satisfy\n\\[\n\\|K_D\\|_{Y_{m,\\rho}\\to Y_{m,\\rho}}=\\frac14,\n\\qquad\n\\|N\\|_{Y_{m,\\rho}\\to B_{m,\\rho}}=\\frac12.\n\\]\nWith the operators extended complex linearly,\n\\[\nN\\Phi_{\\ell,0}=\\frac{e^{im\\ell\\theta}}{2(m|\\ell|+1)},\n\\qquad N\\Phi_{\\ell,s}=0\\quad(s\\ge1).\n\\]",
    "target_statement": "There exist a bounded simply connected domain \\(\\Omega\\subset\\mathbb R^2\\) with \\(C^\\infty\\) boundary, a real number \\(k>0\\), and a nonzero real-valued function \\(u\\in C^2(\\overline\\Omega)\\), such that \\(\\Omega\\) is not a disk and\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
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  "EP-1-H2-to-H3": {
    "id": "EP-1-H2-to-H3",
    "problem_id": "EP-1",
    "problem_name": "Berenstein rigidity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). Let \\(\\phi\\) be holomorphic on a neighborhood of \\(\\overline D\\), with\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D).\n\\]\nSuppose \\(k>0\\) and a real-valued function \\(U\\in C^2(\\overline D)\\) satisfy\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.\n\nThen \\(\\phi\\) is injective on a neighborhood of \\(\\overline D\\). The set \\(\\Omega=\\phi(D)\\) is a bounded simply connected domain with real-analytic Jordan boundary, and\n\\[\nu=U\\circ\\phi^{-1}\n\\]\nis a nonzero real-valued function in \\(C^2(\\overline\\Omega)\\) satisfying\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.\n\nProve H3: For each integer \\(m\\ge2\\) and each \\(\\rho>1\\), define the following spaces and operators. Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor \\(g\\in Y_{m,\\rho}\\), \\(k\\in\\mathbb R\\), and \\(a\\in P_{m,\\rho}\\) with \\(a_0=1\\), define\n\\[\nF_{m,\\rho}(g,k,a)=\\bigl(g+k^2|a|^2K_Dg,\\ (Ng)^2-|a|^2|_{\\partial D}\\bigr).\n\\]\nIts affine domain is\n\\[\nX_{m,\\rho}=\\{(g,k,a)\\in Y_{m,\\rho}\\times\\mathbb R\\times P_{m,\\rho}:a_0=1\\},\n\\]\nwith tangent space \\(X_{m,\\rho}^{\\rm tan}=Y_{m,\\rho}\\times\\mathbb R\\times\\{\\dot a\\in P_{m,\\rho}:\\dot a_0=0\\}\\) and norm\n\\[\n\\|(\\dot g,\\dot k,\\dot a)\\|_X=\\|\\dot g\\|_\\rho+|\\dot k|+\\|\\dot a\\|_P.\n\\]\nIts codomain is \\(Z_{m,\\rho}=Y_{m,\\rho}\\times B_{m,\\rho}\\), with norm\n\\(\\|(y,b)\\|_Z=\\|y\\|_\\rho+\\|b\\|_B\\).\n\nThere exist an integer \\(m\\ge2\\), a real number \\(\\rho>1\\), a point \\(x_c=(g_c,k_c,a_c)\\in X_{m,\\rho}\\) with \\(g_c\\) and \\(a_c\\) of finite coefficient support, a bounded injective real-linear operator\n\\[\nA:Z_{m,\\rho}\\longrightarrow X_{m,\\rho}^{\\rm tan},\n\\]\nand real numbers \\(r>0\\) and \\(0\\le q<1\\) with the following properties.\nWrite\n\\[\nB_r(x_c)=\\{x\\in X_{m,\\rho}:\\|x-x_c\\|_X\\le r\\}.\n\\]\nThe map \\(F_{m,\\rho}\\) is well-defined and continuously Fréchet differentiable on an open neighborhood of \\(B_r(x_c)\\), and\n\\[\n\\|A F_{m,\\rho}(x_c)\\|_X<(1-q)r,\n\\qquad\n\\sup_{x\\in B_r(x_c)}\n\\|I-A\\,DF_{m,\\rho}(x)\\|_{X_{m,\\rho}^{\\rm tan}\\to X_{m,\\rho}^{\\rm tan}}\\le q,\n\\]\nwhere \\(I\\) is the identity on \\(X_{m,\\rho}^{\\rm tan}\\).\nFor every \\((g,k,a)\\in B_r(x_c)\\),\n\\[\nk>0,\\qquad\n\\operatorname{Re}a(z)>0\\quad(z\\in\\overline D),\\qquad\nNg(e^{i\\theta})>0\\quad(\\theta\\in\\mathbb R),\\qquad a_1\\ne0.\n\\]",
    "starting_statement": "Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). Let \\(\\phi\\) be holomorphic on a neighborhood of \\(\\overline D\\), with\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D).\n\\]\nSuppose \\(k>0\\) and a real-valued function \\(U\\in C^2(\\overline D)\\) satisfy\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.\n\nThen \\(\\phi\\) is injective on a neighborhood of \\(\\overline D\\). The set \\(\\Omega=\\phi(D)\\) is a bounded simply connected domain with real-analytic Jordan boundary, and\n\\[\nu=U\\circ\\phi^{-1}\n\\]\nis a nonzero real-valued function in \\(C^2(\\overline\\Omega)\\) satisfying\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
    "target_statement": "For each integer \\(m\\ge2\\) and each \\(\\rho>1\\), define the following spaces and operators. Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor \\(g\\in Y_{m,\\rho}\\), \\(k\\in\\mathbb R\\), and \\(a\\in P_{m,\\rho}\\) with \\(a_0=1\\), define\n\\[\nF_{m,\\rho}(g,k,a)=\\bigl(g+k^2|a|^2K_Dg,\\ (Ng)^2-|a|^2|_{\\partial D}\\bigr).\n\\]\nIts affine domain is\n\\[\nX_{m,\\rho}=\\{(g,k,a)\\in Y_{m,\\rho}\\times\\mathbb R\\times P_{m,\\rho}:a_0=1\\},\n\\]\nwith tangent space \\(X_{m,\\rho}^{\\rm tan}=Y_{m,\\rho}\\times\\mathbb R\\times\\{\\dot a\\in P_{m,\\rho}:\\dot a_0=0\\}\\) and norm\n\\[\n\\|(\\dot g,\\dot k,\\dot a)\\|_X=\\|\\dot g\\|_\\rho+|\\dot k|+\\|\\dot a\\|_P.\n\\]\nIts codomain is \\(Z_{m,\\rho}=Y_{m,\\rho}\\times B_{m,\\rho}\\), with norm\n\\(\\|(y,b)\\|_Z=\\|y\\|_\\rho+\\|b\\|_B\\).\n\nThere exist an integer \\(m\\ge2\\), a real number \\(\\rho>1\\), a point \\(x_c=(g_c,k_c,a_c)\\in X_{m,\\rho}\\) with \\(g_c\\) and \\(a_c\\) of finite coefficient support, a bounded injective real-linear operator\n\\[\nA:Z_{m,\\rho}\\longrightarrow X_{m,\\rho}^{\\rm tan},\n\\]\nand real numbers \\(r>0\\) and \\(0\\le q<1\\) with the following properties.\nWrite\n\\[\nB_r(x_c)=\\{x\\in X_{m,\\rho}:\\|x-x_c\\|_X\\le r\\}.\n\\]\nThe map \\(F_{m,\\rho}\\) is well-defined and continuously Fréchet differentiable on an open neighborhood of \\(B_r(x_c)\\), and\n\\[\n\\|A F_{m,\\rho}(x_c)\\|_X<(1-q)r,\n\\qquad\n\\sup_{x\\in B_r(x_c)}\n\\|I-A\\,DF_{m,\\rho}(x)\\|_{X_{m,\\rho}^{\\rm tan}\\to X_{m,\\rho}^{\\rm tan}}\\le q,\n\\]\nwhere \\(I\\) is the identity on \\(X_{m,\\rho}^{\\rm tan}\\).\nFor every \\((g,k,a)\\in B_r(x_c)\\),\n\\[\nk>0,\\qquad\n\\operatorname{Re}a(z)>0\\quad(z\\in\\overline D),\\qquad\nNg(e^{i\\theta})>0\\quad(\\theta\\in\\mathbb R),\\qquad a_1\\ne0.\n\\]",
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  "EP-1-H2-to-H4": {
    "id": "EP-1-H2-to-H4",
    "problem_id": "EP-1",
    "problem_name": "Berenstein rigidity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). Let \\(\\phi\\) be holomorphic on a neighborhood of \\(\\overline D\\), with\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D).\n\\]\nSuppose \\(k>0\\) and a real-valued function \\(U\\in C^2(\\overline D)\\) satisfy\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.\n\nThen \\(\\phi\\) is injective on a neighborhood of \\(\\overline D\\). The set \\(\\Omega=\\phi(D)\\) is a bounded simply connected domain with real-analytic Jordan boundary, and\n\\[\nu=U\\circ\\phi^{-1}\n\\]\nis a nonzero real-valued function in \\(C^2(\\overline\\Omega)\\) satisfying\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.\n\nProve H4: Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). There exist an integer \\(m\\ge2\\), a real number \\(k>0\\), a holomorphic function \\(\\phi\\) on a neighborhood of \\(\\overline D\\), and a real-valued function \\(U\\in C^2(\\overline D)\\), such that\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D),\n\\]\n\\[\n\\phi(e^{2\\pi i/m}z)=e^{2\\pi i/m}\\phi(z)\\quad(z\\in D),\n\\qquad \\phi\\not\\equiv z,\n\\]\nand\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.",
    "starting_statement": "Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). Let \\(\\phi\\) be holomorphic on a neighborhood of \\(\\overline D\\), with\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D).\n\\]\nSuppose \\(k>0\\) and a real-valued function \\(U\\in C^2(\\overline D)\\) satisfy\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.\n\nThen \\(\\phi\\) is injective on a neighborhood of \\(\\overline D\\). The set \\(\\Omega=\\phi(D)\\) is a bounded simply connected domain with real-analytic Jordan boundary, and\n\\[\nu=U\\circ\\phi^{-1}\n\\]\nis a nonzero real-valued function in \\(C^2(\\overline\\Omega)\\) satisfying\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
    "target_statement": "Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). There exist an integer \\(m\\ge2\\), a real number \\(k>0\\), a holomorphic function \\(\\phi\\) on a neighborhood of \\(\\overline D\\), and a real-valued function \\(U\\in C^2(\\overline D)\\), such that\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D),\n\\]\n\\[\n\\phi(e^{2\\pi i/m}z)=e^{2\\pi i/m}\\phi(z)\\quad(z\\in D),\n\\qquad \\phi\\not\\equiv z,\n\\]\nand\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.",
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  "EP-1-H2-to-H5": {
    "id": "EP-1-H2-to-H5",
    "problem_id": "EP-1",
    "problem_name": "Berenstein rigidity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). Let \\(\\phi\\) be holomorphic on a neighborhood of \\(\\overline D\\), with\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D).\n\\]\nSuppose \\(k>0\\) and a real-valued function \\(U\\in C^2(\\overline D)\\) satisfy\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.\n\nThen \\(\\phi\\) is injective on a neighborhood of \\(\\overline D\\). The set \\(\\Omega=\\phi(D)\\) is a bounded simply connected domain with real-analytic Jordan boundary, and\n\\[\nu=U\\circ\\phi^{-1}\n\\]\nis a nonzero real-valued function in \\(C^2(\\overline\\Omega)\\) satisfying\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.\n\nProve H5: There exist a bounded simply connected domain \\(\\Omega\\subset\\mathbb R^2\\) with \\(C^\\infty\\) boundary, a real number \\(k>0\\), and a nonzero real-valued function \\(u\\in C^2(\\overline\\Omega)\\), such that \\(\\Omega\\) is not a disk and\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
    "starting_statement": "Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). Let \\(\\phi\\) be holomorphic on a neighborhood of \\(\\overline D\\), with\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D).\n\\]\nSuppose \\(k>0\\) and a real-valued function \\(U\\in C^2(\\overline D)\\) satisfy\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.\n\nThen \\(\\phi\\) is injective on a neighborhood of \\(\\overline D\\). The set \\(\\Omega=\\phi(D)\\) is a bounded simply connected domain with real-analytic Jordan boundary, and\n\\[\nu=U\\circ\\phi^{-1}\n\\]\nis a nonzero real-valued function in \\(C^2(\\overline\\Omega)\\) satisfying\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
    "target_statement": "There exist a bounded simply connected domain \\(\\Omega\\subset\\mathbb R^2\\) with \\(C^\\infty\\) boundary, a real number \\(k>0\\), and a nonzero real-valued function \\(u\\in C^2(\\overline\\Omega)\\), such that \\(\\Omega\\) is not a disk and\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
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  "EP-1-H3-to-H4": {
    "id": "EP-1-H3-to-H4",
    "problem_id": "EP-1",
    "problem_name": "Berenstein rigidity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H3: For each integer \\(m\\ge2\\) and each \\(\\rho>1\\), define the following spaces and operators. Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor \\(g\\in Y_{m,\\rho}\\), \\(k\\in\\mathbb R\\), and \\(a\\in P_{m,\\rho}\\) with \\(a_0=1\\), define\n\\[\nF_{m,\\rho}(g,k,a)=\\bigl(g+k^2|a|^2K_Dg,\\ (Ng)^2-|a|^2|_{\\partial D}\\bigr).\n\\]\nIts affine domain is\n\\[\nX_{m,\\rho}=\\{(g,k,a)\\in Y_{m,\\rho}\\times\\mathbb R\\times P_{m,\\rho}:a_0=1\\},\n\\]\nwith tangent space \\(X_{m,\\rho}^{\\rm tan}=Y_{m,\\rho}\\times\\mathbb R\\times\\{\\dot a\\in P_{m,\\rho}:\\dot a_0=0\\}\\) and norm\n\\[\n\\|(\\dot g,\\dot k,\\dot a)\\|_X=\\|\\dot g\\|_\\rho+|\\dot k|+\\|\\dot a\\|_P.\n\\]\nIts codomain is \\(Z_{m,\\rho}=Y_{m,\\rho}\\times B_{m,\\rho}\\), with norm\n\\(\\|(y,b)\\|_Z=\\|y\\|_\\rho+\\|b\\|_B\\).\n\nThere exist an integer \\(m\\ge2\\), a real number \\(\\rho>1\\), a point \\(x_c=(g_c,k_c,a_c)\\in X_{m,\\rho}\\) with \\(g_c\\) and \\(a_c\\) of finite coefficient support, a bounded injective real-linear operator\n\\[\nA:Z_{m,\\rho}\\longrightarrow X_{m,\\rho}^{\\rm tan},\n\\]\nand real numbers \\(r>0\\) and \\(0\\le q<1\\) with the following properties.\nWrite\n\\[\nB_r(x_c)=\\{x\\in X_{m,\\rho}:\\|x-x_c\\|_X\\le r\\}.\n\\]\nThe map \\(F_{m,\\rho}\\) is well-defined and continuously Fréchet differentiable on an open neighborhood of \\(B_r(x_c)\\), and\n\\[\n\\|A F_{m,\\rho}(x_c)\\|_X<(1-q)r,\n\\qquad\n\\sup_{x\\in B_r(x_c)}\n\\|I-A\\,DF_{m,\\rho}(x)\\|_{X_{m,\\rho}^{\\rm tan}\\to X_{m,\\rho}^{\\rm tan}}\\le q,\n\\]\nwhere \\(I\\) is the identity on \\(X_{m,\\rho}^{\\rm tan}\\).\nFor every \\((g,k,a)\\in B_r(x_c)\\),\n\\[\nk>0,\\qquad\n\\operatorname{Re}a(z)>0\\quad(z\\in\\overline D),\\qquad\nNg(e^{i\\theta})>0\\quad(\\theta\\in\\mathbb R),\\qquad a_1\\ne0.\n\\]\n\nProve H4: Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). There exist an integer \\(m\\ge2\\), a real number \\(k>0\\), a holomorphic function \\(\\phi\\) on a neighborhood of \\(\\overline D\\), and a real-valued function \\(U\\in C^2(\\overline D)\\), such that\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D),\n\\]\n\\[\n\\phi(e^{2\\pi i/m}z)=e^{2\\pi i/m}\\phi(z)\\quad(z\\in D),\n\\qquad \\phi\\not\\equiv z,\n\\]\nand\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.",
    "starting_statement": "For each integer \\(m\\ge2\\) and each \\(\\rho>1\\), define the following spaces and operators. Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor \\(g\\in Y_{m,\\rho}\\), \\(k\\in\\mathbb R\\), and \\(a\\in P_{m,\\rho}\\) with \\(a_0=1\\), define\n\\[\nF_{m,\\rho}(g,k,a)=\\bigl(g+k^2|a|^2K_Dg,\\ (Ng)^2-|a|^2|_{\\partial D}\\bigr).\n\\]\nIts affine domain is\n\\[\nX_{m,\\rho}=\\{(g,k,a)\\in Y_{m,\\rho}\\times\\mathbb R\\times P_{m,\\rho}:a_0=1\\},\n\\]\nwith tangent space \\(X_{m,\\rho}^{\\rm tan}=Y_{m,\\rho}\\times\\mathbb R\\times\\{\\dot a\\in P_{m,\\rho}:\\dot a_0=0\\}\\) and norm\n\\[\n\\|(\\dot g,\\dot k,\\dot a)\\|_X=\\|\\dot g\\|_\\rho+|\\dot k|+\\|\\dot a\\|_P.\n\\]\nIts codomain is \\(Z_{m,\\rho}=Y_{m,\\rho}\\times B_{m,\\rho}\\), with norm\n\\(\\|(y,b)\\|_Z=\\|y\\|_\\rho+\\|b\\|_B\\).\n\nThere exist an integer \\(m\\ge2\\), a real number \\(\\rho>1\\), a point \\(x_c=(g_c,k_c,a_c)\\in X_{m,\\rho}\\) with \\(g_c\\) and \\(a_c\\) of finite coefficient support, a bounded injective real-linear operator\n\\[\nA:Z_{m,\\rho}\\longrightarrow X_{m,\\rho}^{\\rm tan},\n\\]\nand real numbers \\(r>0\\) and \\(0\\le q<1\\) with the following properties.\nWrite\n\\[\nB_r(x_c)=\\{x\\in X_{m,\\rho}:\\|x-x_c\\|_X\\le r\\}.\n\\]\nThe map \\(F_{m,\\rho}\\) is well-defined and continuously Fréchet differentiable on an open neighborhood of \\(B_r(x_c)\\), and\n\\[\n\\|A F_{m,\\rho}(x_c)\\|_X<(1-q)r,\n\\qquad\n\\sup_{x\\in B_r(x_c)}\n\\|I-A\\,DF_{m,\\rho}(x)\\|_{X_{m,\\rho}^{\\rm tan}\\to X_{m,\\rho}^{\\rm tan}}\\le q,\n\\]\nwhere \\(I\\) is the identity on \\(X_{m,\\rho}^{\\rm tan}\\).\nFor every \\((g,k,a)\\in B_r(x_c)\\),\n\\[\nk>0,\\qquad\n\\operatorname{Re}a(z)>0\\quad(z\\in\\overline D),\\qquad\nNg(e^{i\\theta})>0\\quad(\\theta\\in\\mathbb R),\\qquad a_1\\ne0.\n\\]",
    "target_statement": "Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). There exist an integer \\(m\\ge2\\), a real number \\(k>0\\), a holomorphic function \\(\\phi\\) on a neighborhood of \\(\\overline D\\), and a real-valued function \\(U\\in C^2(\\overline D)\\), such that\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D),\n\\]\n\\[\n\\phi(e^{2\\pi i/m}z)=e^{2\\pi i/m}\\phi(z)\\quad(z\\in D),\n\\qquad \\phi\\not\\equiv z,\n\\]\nand\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.",
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  "EP-1-H3-to-H5": {
    "id": "EP-1-H3-to-H5",
    "problem_id": "EP-1",
    "problem_name": "Berenstein rigidity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H3: For each integer \\(m\\ge2\\) and each \\(\\rho>1\\), define the following spaces and operators. Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor \\(g\\in Y_{m,\\rho}\\), \\(k\\in\\mathbb R\\), and \\(a\\in P_{m,\\rho}\\) with \\(a_0=1\\), define\n\\[\nF_{m,\\rho}(g,k,a)=\\bigl(g+k^2|a|^2K_Dg,\\ (Ng)^2-|a|^2|_{\\partial D}\\bigr).\n\\]\nIts affine domain is\n\\[\nX_{m,\\rho}=\\{(g,k,a)\\in Y_{m,\\rho}\\times\\mathbb R\\times P_{m,\\rho}:a_0=1\\},\n\\]\nwith tangent space \\(X_{m,\\rho}^{\\rm tan}=Y_{m,\\rho}\\times\\mathbb R\\times\\{\\dot a\\in P_{m,\\rho}:\\dot a_0=0\\}\\) and norm\n\\[\n\\|(\\dot g,\\dot k,\\dot a)\\|_X=\\|\\dot g\\|_\\rho+|\\dot k|+\\|\\dot a\\|_P.\n\\]\nIts codomain is \\(Z_{m,\\rho}=Y_{m,\\rho}\\times B_{m,\\rho}\\), with norm\n\\(\\|(y,b)\\|_Z=\\|y\\|_\\rho+\\|b\\|_B\\).\n\nThere exist an integer \\(m\\ge2\\), a real number \\(\\rho>1\\), a point \\(x_c=(g_c,k_c,a_c)\\in X_{m,\\rho}\\) with \\(g_c\\) and \\(a_c\\) of finite coefficient support, a bounded injective real-linear operator\n\\[\nA:Z_{m,\\rho}\\longrightarrow X_{m,\\rho}^{\\rm tan},\n\\]\nand real numbers \\(r>0\\) and \\(0\\le q<1\\) with the following properties.\nWrite\n\\[\nB_r(x_c)=\\{x\\in X_{m,\\rho}:\\|x-x_c\\|_X\\le r\\}.\n\\]\nThe map \\(F_{m,\\rho}\\) is well-defined and continuously Fréchet differentiable on an open neighborhood of \\(B_r(x_c)\\), and\n\\[\n\\|A F_{m,\\rho}(x_c)\\|_X<(1-q)r,\n\\qquad\n\\sup_{x\\in B_r(x_c)}\n\\|I-A\\,DF_{m,\\rho}(x)\\|_{X_{m,\\rho}^{\\rm tan}\\to X_{m,\\rho}^{\\rm tan}}\\le q,\n\\]\nwhere \\(I\\) is the identity on \\(X_{m,\\rho}^{\\rm tan}\\).\nFor every \\((g,k,a)\\in B_r(x_c)\\),\n\\[\nk>0,\\qquad\n\\operatorname{Re}a(z)>0\\quad(z\\in\\overline D),\\qquad\nNg(e^{i\\theta})>0\\quad(\\theta\\in\\mathbb R),\\qquad a_1\\ne0.\n\\]\n\nProve H5: There exist a bounded simply connected domain \\(\\Omega\\subset\\mathbb R^2\\) with \\(C^\\infty\\) boundary, a real number \\(k>0\\), and a nonzero real-valued function \\(u\\in C^2(\\overline\\Omega)\\), such that \\(\\Omega\\) is not a disk and\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
    "starting_statement": "For each integer \\(m\\ge2\\) and each \\(\\rho>1\\), define the following spaces and operators. Put \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). For \\(\\ell\\in\\mathbb Z\\) and \\(s\\in\\mathbb Z_{\\ge0}\\), set\n\\[\n\\Phi_{\\ell,s}(r,\\theta)=r^{m|\\ell|}P_s^{(0,m|\\ell|)}(2r^2-1)e^{im\\ell\\theta},\n\\]\nwhere \\(P_s^{(0,b)}\\) is the Jacobi polynomial normalized by \\(P_s^{(0,b)}(1)=1\\).\nLet \\(W_{m,\\rho}\\) consist of complex coefficient series \\(g=\\sum_{\\ell,s}g_{\\ell,s}\\Phi_{\\ell,s}\\) with\n\\[\n\\|g\\|_\\rho=\\sum_{\\ell,s}\\rho^{|\\ell|}|g_{\\ell,s}|<\\infty.\n\\]\nLet \\(Y_{m,\\rho}\\) be the real subspace of \\(W_{m,\\rho}\\) consisting of real-valued functions invariant under \\(\\theta\\mapsto-\\theta\\).\nLet \\(P_{m,\\rho}\\) consist of holomorphic series \\(a(z)=\\sum_{j\\ge0}a_jz^{mj}\\) with real coefficients and norm\n\\(\\|a\\|_P=\\sum_{j\\ge0}\\rho^j|a_j|<\\infty\\).\nLet \\(B_{m,\\rho}\\) consist of real-valued, reflection-invariant boundary series\n\\(b(\\theta)=\\sum_{\\ell\\in\\mathbb Z}b_\\ell e^{im\\ell\\theta}\\), with norm\n\\(\\|b\\|_B=\\sum_\\ell\\rho^{|\\ell|}|b_\\ell|<\\infty\\).\nFor \\(g\\in Y_{m,\\rho}\\), let \\(K_Dg\\) denote the weak solution of \\(\\Delta U=g\\) in \\(D\\) with zero Dirichlet trace; put \\(Ng=\\partial_r(K_Dg)|_{\\partial D}\\), with the derivative understood as its boundary trace.\n\nFor \\(g\\in Y_{m,\\rho}\\), \\(k\\in\\mathbb R\\), and \\(a\\in P_{m,\\rho}\\) with \\(a_0=1\\), define\n\\[\nF_{m,\\rho}(g,k,a)=\\bigl(g+k^2|a|^2K_Dg,\\ (Ng)^2-|a|^2|_{\\partial D}\\bigr).\n\\]\nIts affine domain is\n\\[\nX_{m,\\rho}=\\{(g,k,a)\\in Y_{m,\\rho}\\times\\mathbb R\\times P_{m,\\rho}:a_0=1\\},\n\\]\nwith tangent space \\(X_{m,\\rho}^{\\rm tan}=Y_{m,\\rho}\\times\\mathbb R\\times\\{\\dot a\\in P_{m,\\rho}:\\dot a_0=0\\}\\) and norm\n\\[\n\\|(\\dot g,\\dot k,\\dot a)\\|_X=\\|\\dot g\\|_\\rho+|\\dot k|+\\|\\dot a\\|_P.\n\\]\nIts codomain is \\(Z_{m,\\rho}=Y_{m,\\rho}\\times B_{m,\\rho}\\), with norm\n\\(\\|(y,b)\\|_Z=\\|y\\|_\\rho+\\|b\\|_B\\).\n\nThere exist an integer \\(m\\ge2\\), a real number \\(\\rho>1\\), a point \\(x_c=(g_c,k_c,a_c)\\in X_{m,\\rho}\\) with \\(g_c\\) and \\(a_c\\) of finite coefficient support, a bounded injective real-linear operator\n\\[\nA:Z_{m,\\rho}\\longrightarrow X_{m,\\rho}^{\\rm tan},\n\\]\nand real numbers \\(r>0\\) and \\(0\\le q<1\\) with the following properties.\nWrite\n\\[\nB_r(x_c)=\\{x\\in X_{m,\\rho}:\\|x-x_c\\|_X\\le r\\}.\n\\]\nThe map \\(F_{m,\\rho}\\) is well-defined and continuously Fréchet differentiable on an open neighborhood of \\(B_r(x_c)\\), and\n\\[\n\\|A F_{m,\\rho}(x_c)\\|_X<(1-q)r,\n\\qquad\n\\sup_{x\\in B_r(x_c)}\n\\|I-A\\,DF_{m,\\rho}(x)\\|_{X_{m,\\rho}^{\\rm tan}\\to X_{m,\\rho}^{\\rm tan}}\\le q,\n\\]\nwhere \\(I\\) is the identity on \\(X_{m,\\rho}^{\\rm tan}\\).\nFor every \\((g,k,a)\\in B_r(x_c)\\),\n\\[\nk>0,\\qquad\n\\operatorname{Re}a(z)>0\\quad(z\\in\\overline D),\\qquad\nNg(e^{i\\theta})>0\\quad(\\theta\\in\\mathbb R),\\qquad a_1\\ne0.\n\\]",
    "target_statement": "There exist a bounded simply connected domain \\(\\Omega\\subset\\mathbb R^2\\) with \\(C^\\infty\\) boundary, a real number \\(k>0\\), and a nonzero real-valued function \\(u\\in C^2(\\overline\\Omega)\\), such that \\(\\Omega\\) is not a disk and\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
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  "EP-1-H4-to-H5": {
    "id": "EP-1-H4-to-H5",
    "problem_id": "EP-1",
    "problem_name": "Berenstein rigidity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 4,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H4: Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). There exist an integer \\(m\\ge2\\), a real number \\(k>0\\), a holomorphic function \\(\\phi\\) on a neighborhood of \\(\\overline D\\), and a real-valued function \\(U\\in C^2(\\overline D)\\), such that\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D),\n\\]\n\\[\n\\phi(e^{2\\pi i/m}z)=e^{2\\pi i/m}\\phi(z)\\quad(z\\in D),\n\\qquad \\phi\\not\\equiv z,\n\\]\nand\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.\n\nProve H5: There exist a bounded simply connected domain \\(\\Omega\\subset\\mathbb R^2\\) with \\(C^\\infty\\) boundary, a real number \\(k>0\\), and a nonzero real-valued function \\(u\\in C^2(\\overline\\Omega)\\), such that \\(\\Omega\\) is not a disk and\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
    "starting_statement": "Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). There exist an integer \\(m\\ge2\\), a real number \\(k>0\\), a holomorphic function \\(\\phi\\) on a neighborhood of \\(\\overline D\\), and a real-valued function \\(U\\in C^2(\\overline D)\\), such that\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D),\n\\]\n\\[\n\\phi(e^{2\\pi i/m}z)=e^{2\\pi i/m}\\phi(z)\\quad(z\\in D),\n\\qquad \\phi\\not\\equiv z,\n\\]\nand\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.",
    "target_statement": "There exist a bounded simply connected domain \\(\\Omega\\subset\\mathbb R^2\\) with \\(C^\\infty\\) boundary, a real number \\(k>0\\), and a nonzero real-valued function \\(u\\in C^2(\\overline\\Omega)\\), such that \\(\\Omega\\) is not a disk and\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
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  "KT-1-H0-to-H1": {
    "id": "KT-1-H0-to-H1",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 1,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H1: For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).",
    "starting_statement": null,
    "target_statement": "For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).",
    "source_hashes": {
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  "KT-1-H0-to-H2": {
    "id": "KT-1-H0-to-H2",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 2,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H2: Let \\(K\\subset S^3\\) be a smooth oriented knot admitting an orientation-reversing diffeomorphism \\(h:S^3\\to S^3\\) with \\(h^2=\\operatorname{id}\\), \\(h(K)=K\\), and \\(h|_K\\) preserving orientation.\nThere exists \\(\\varphi\\in\\mathbb Z[w]\\) such that\n\\[\n\\nabla_K(z)=\\varphi(z^2)^2.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\).",
    "starting_statement": null,
    "target_statement": "Let \\(K\\subset S^3\\) be a smooth oriented knot admitting an orientation-reversing diffeomorphism \\(h:S^3\\to S^3\\) with \\(h^2=\\operatorname{id}\\), \\(h(K)=K\\), and \\(h|_K\\) preserving orientation.\nThere exists \\(\\varphi\\in\\mathbb Z[w]\\) such that\n\\[\n\\nabla_K(z)=\\varphi(z^2)^2.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\).",
    "source_hashes": {
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  "KT-1-H0-to-H3": {
    "id": "KT-1-H0-to-H3",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H3: Let \\(L=K\\cup C\\subset S^3\\) be a smooth oriented two-component link with \\(C\\) unknotted and \\(\\operatorname{lk}(K,C)\\) odd. Let \\(\\pi:S^3\\to S^3\\) be the oriented double cover branched over \\(C\\), and give \\(K'=\\pi^{-1}(K)\\) and \\(C'=\\pi^{-1}(C)\\) the lifted orientations.\nThen \\(K'\\) is a knot and\n\\[\n\\Delta_{K'\\cup C'}(t,y^2)\\doteq\\Delta_{K\\cup C}(t,y)\\Delta_{K\\cup C}(t,-y).\n\\]\nThe variables \\(t,y\\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \\(\\doteq\\) means equality up to multiplication by \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
    "starting_statement": null,
    "target_statement": "Let \\(L=K\\cup C\\subset S^3\\) be a smooth oriented two-component link with \\(C\\) unknotted and \\(\\operatorname{lk}(K,C)\\) odd. Let \\(\\pi:S^3\\to S^3\\) be the oriented double cover branched over \\(C\\), and give \\(K'=\\pi^{-1}(K)\\) and \\(C'=\\pi^{-1}(C)\\) the lifted orientations.\nThen \\(K'\\) is a knot and\n\\[\n\\Delta_{K'\\cup C'}(t,y^2)\\doteq\\Delta_{K\\cup C}(t,y)\\Delta_{K\\cup C}(t,-y).\n\\]\nThe variables \\(t,y\\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \\(\\doteq\\) means equality up to multiplication by \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
    "source_hashes": {
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  },
  "KT-1-H0-to-H4": {
    "id": "KT-1-H0-to-H4",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H4: Identify \\(S^3\\) with \\(\\{(z_1,z_2)\\in\\mathbb C^2:|z_1|^2+|z_2|^2=1\\}\\). For integers \\(a\\ge2\\) and odd \\(c\\), put\n\\[\n\\rho(z_1,z_2)=(e^{2\\pi ic/2^a}z_1,\\overline{z_2}),\\qquad C=\\{z_1=0\\}.\n\\]\nLet \\(K\\subset S^3\\) be a smooth oriented knot disjoint from \\(C\\), with \\(\\rho(K)=K\\), \\(\\rho|_K\\) preserving orientation, and \\(\\operatorname{lk}(K,C)\\) odd.\nThere exists \\(Q\\in\\mathbb Z[U,V]\\) such that\n\\[\n\\Delta_{K\\cup C}(t,y)\\doteq Q(t+t^{-1},y+y^{-1}).\n\\]\nThe variables \\(t,y\\) correspond to \\(K,C\\), respectively, and \\(\\doteq\\) denotes equality up to a Laurent unit \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
    "starting_statement": null,
    "target_statement": "Identify \\(S^3\\) with \\(\\{(z_1,z_2)\\in\\mathbb C^2:|z_1|^2+|z_2|^2=1\\}\\). For integers \\(a\\ge2\\) and odd \\(c\\), put\n\\[\n\\rho(z_1,z_2)=(e^{2\\pi ic/2^a}z_1,\\overline{z_2}),\\qquad C=\\{z_1=0\\}.\n\\]\nLet \\(K\\subset S^3\\) be a smooth oriented knot disjoint from \\(C\\), with \\(\\rho(K)=K\\), \\(\\rho|_K\\) preserving orientation, and \\(\\operatorname{lk}(K,C)\\) odd.\nThere exists \\(Q\\in\\mathbb Z[U,V]\\) such that\n\\[\n\\Delta_{K\\cup C}(t,y)\\doteq Q(t+t^{-1},y+y^{-1}).\n\\]\nThe variables \\(t,y\\) correspond to \\(K,C\\), respectively, and \\(\\doteq\\) denotes equality up to a Laurent unit \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
    "source_hashes": {
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  },
  "KT-1-H0-to-H5": {
    "id": "KT-1-H0-to-H5",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H5: Let \\(K\\subset S^3\\) be an oriented knot admitting a finite-order orientation-reversing diffeomorphism of \\(S^3\\) that preserves the orientation of \\(K\\). Let \\(\\nabla_K(z)\\in\\mathbb Z[z]\\) be its Conway polynomial, normalized by \\(\\nabla_{\\rm unknot}=1\\).\nThere exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4,\n\\]\nwhere the congruence is coefficientwise.",
    "starting_statement": null,
    "target_statement": "Let \\(K\\subset S^3\\) be an oriented knot admitting a finite-order orientation-reversing diffeomorphism of \\(S^3\\) that preserves the orientation of \\(K\\). Let \\(\\nabla_K(z)\\in\\mathbb Z[z]\\) be its Conway polynomial, normalized by \\(\\nabla_{\\rm unknot}=1\\).\nThere exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4,\n\\]\nwhere the congruence is coefficientwise.",
    "source_hashes": {
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  "KT-1-H0-to-H6": {
    "id": "KT-1-H0-to-H6",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H6: A knot \\(K\\subset S^3\\) is called amphicheiral if an orientation-reversing homeomorphism of \\(S^3\\) maps \\(K\\) to itself; the homeomorphism may preserve or reverse the knot orientation.\nFor every amphicheiral knot \\(K\\), including the unknot, there exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\), and the congruence is coefficientwise.",
    "starting_statement": null,
    "target_statement": "A knot \\(K\\subset S^3\\) is called amphicheiral if an orientation-reversing homeomorphism of \\(S^3\\) maps \\(K\\) to itself; the homeomorphism may preserve or reverse the knot orientation.\nFor every amphicheiral knot \\(K\\), including the unknot, there exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\), and the congruence is coefficientwise.",
    "source_hashes": {
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  "KT-1-H1-to-H2": {
    "id": "KT-1-H1-to-H2",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 2,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).\n\nProve H2: Let \\(K\\subset S^3\\) be a smooth oriented knot admitting an orientation-reversing diffeomorphism \\(h:S^3\\to S^3\\) with \\(h^2=\\operatorname{id}\\), \\(h(K)=K\\), and \\(h|_K\\) preserving orientation.\nThere exists \\(\\varphi\\in\\mathbb Z[w]\\) such that\n\\[\n\\nabla_K(z)=\\varphi(z^2)^2.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\).",
    "starting_statement": "For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).",
    "target_statement": "Let \\(K\\subset S^3\\) be a smooth oriented knot admitting an orientation-reversing diffeomorphism \\(h:S^3\\to S^3\\) with \\(h^2=\\operatorname{id}\\), \\(h(K)=K\\), and \\(h|_K\\) preserving orientation.\nThere exists \\(\\varphi\\in\\mathbb Z[w]\\) such that\n\\[\n\\nabla_K(z)=\\varphi(z^2)^2.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\).",
    "source_hashes": {
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  },
  "KT-1-H1-to-H3": {
    "id": "KT-1-H1-to-H3",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).\n\nProve H3: Let \\(L=K\\cup C\\subset S^3\\) be a smooth oriented two-component link with \\(C\\) unknotted and \\(\\operatorname{lk}(K,C)\\) odd. Let \\(\\pi:S^3\\to S^3\\) be the oriented double cover branched over \\(C\\), and give \\(K'=\\pi^{-1}(K)\\) and \\(C'=\\pi^{-1}(C)\\) the lifted orientations.\nThen \\(K'\\) is a knot and\n\\[\n\\Delta_{K'\\cup C'}(t,y^2)\\doteq\\Delta_{K\\cup C}(t,y)\\Delta_{K\\cup C}(t,-y).\n\\]\nThe variables \\(t,y\\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \\(\\doteq\\) means equality up to multiplication by \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
    "starting_statement": "For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).",
    "target_statement": "Let \\(L=K\\cup C\\subset S^3\\) be a smooth oriented two-component link with \\(C\\) unknotted and \\(\\operatorname{lk}(K,C)\\) odd. Let \\(\\pi:S^3\\to S^3\\) be the oriented double cover branched over \\(C\\), and give \\(K'=\\pi^{-1}(K)\\) and \\(C'=\\pi^{-1}(C)\\) the lifted orientations.\nThen \\(K'\\) is a knot and\n\\[\n\\Delta_{K'\\cup C'}(t,y^2)\\doteq\\Delta_{K\\cup C}(t,y)\\Delta_{K\\cup C}(t,-y).\n\\]\nThe variables \\(t,y\\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \\(\\doteq\\) means equality up to multiplication by \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
    "source_hashes": {
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  "KT-1-H1-to-H4": {
    "id": "KT-1-H1-to-H4",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).\n\nProve H4: Identify \\(S^3\\) with \\(\\{(z_1,z_2)\\in\\mathbb C^2:|z_1|^2+|z_2|^2=1\\}\\). For integers \\(a\\ge2\\) and odd \\(c\\), put\n\\[\n\\rho(z_1,z_2)=(e^{2\\pi ic/2^a}z_1,\\overline{z_2}),\\qquad C=\\{z_1=0\\}.\n\\]\nLet \\(K\\subset S^3\\) be a smooth oriented knot disjoint from \\(C\\), with \\(\\rho(K)=K\\), \\(\\rho|_K\\) preserving orientation, and \\(\\operatorname{lk}(K,C)\\) odd.\nThere exists \\(Q\\in\\mathbb Z[U,V]\\) such that\n\\[\n\\Delta_{K\\cup C}(t,y)\\doteq Q(t+t^{-1},y+y^{-1}).\n\\]\nThe variables \\(t,y\\) correspond to \\(K,C\\), respectively, and \\(\\doteq\\) denotes equality up to a Laurent unit \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
    "starting_statement": "For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).",
    "target_statement": "Identify \\(S^3\\) with \\(\\{(z_1,z_2)\\in\\mathbb C^2:|z_1|^2+|z_2|^2=1\\}\\). For integers \\(a\\ge2\\) and odd \\(c\\), put\n\\[\n\\rho(z_1,z_2)=(e^{2\\pi ic/2^a}z_1,\\overline{z_2}),\\qquad C=\\{z_1=0\\}.\n\\]\nLet \\(K\\subset S^3\\) be a smooth oriented knot disjoint from \\(C\\), with \\(\\rho(K)=K\\), \\(\\rho|_K\\) preserving orientation, and \\(\\operatorname{lk}(K,C)\\) odd.\nThere exists \\(Q\\in\\mathbb Z[U,V]\\) such that\n\\[\n\\Delta_{K\\cup C}(t,y)\\doteq Q(t+t^{-1},y+y^{-1}).\n\\]\nThe variables \\(t,y\\) correspond to \\(K,C\\), respectively, and \\(\\doteq\\) denotes equality up to a Laurent unit \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
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  "KT-1-H1-to-H5": {
    "id": "KT-1-H1-to-H5",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).\n\nProve H5: Let \\(K\\subset S^3\\) be an oriented knot admitting a finite-order orientation-reversing diffeomorphism of \\(S^3\\) that preserves the orientation of \\(K\\). Let \\(\\nabla_K(z)\\in\\mathbb Z[z]\\) be its Conway polynomial, normalized by \\(\\nabla_{\\rm unknot}=1\\).\nThere exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4,\n\\]\nwhere the congruence is coefficientwise.",
    "starting_statement": "For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).",
    "target_statement": "Let \\(K\\subset S^3\\) be an oriented knot admitting a finite-order orientation-reversing diffeomorphism of \\(S^3\\) that preserves the orientation of \\(K\\). Let \\(\\nabla_K(z)\\in\\mathbb Z[z]\\) be its Conway polynomial, normalized by \\(\\nabla_{\\rm unknot}=1\\).\nThere exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4,\n\\]\nwhere the congruence is coefficientwise.",
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  "KT-1-H1-to-H6": {
    "id": "KT-1-H1-to-H6",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).\n\nProve H6: A knot \\(K\\subset S^3\\) is called amphicheiral if an orientation-reversing homeomorphism of \\(S^3\\) maps \\(K\\) to itself; the homeomorphism may preserve or reverse the knot orientation.\nFor every amphicheiral knot \\(K\\), including the unknot, there exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\), and the congruence is coefficientwise.",
    "starting_statement": "For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).",
    "target_statement": "A knot \\(K\\subset S^3\\) is called amphicheiral if an orientation-reversing homeomorphism of \\(S^3\\) maps \\(K\\) to itself; the homeomorphism may preserve or reverse the knot orientation.\nFor every amphicheiral knot \\(K\\), including the unknot, there exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\), and the congruence is coefficientwise.",
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  "KT-1-H2-to-H3": {
    "id": "KT-1-H2-to-H3",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(K\\subset S^3\\) be a smooth oriented knot admitting an orientation-reversing diffeomorphism \\(h:S^3\\to S^3\\) with \\(h^2=\\operatorname{id}\\), \\(h(K)=K\\), and \\(h|_K\\) preserving orientation.\nThere exists \\(\\varphi\\in\\mathbb Z[w]\\) such that\n\\[\n\\nabla_K(z)=\\varphi(z^2)^2.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\).\n\nProve H3: Let \\(L=K\\cup C\\subset S^3\\) be a smooth oriented two-component link with \\(C\\) unknotted and \\(\\operatorname{lk}(K,C)\\) odd. Let \\(\\pi:S^3\\to S^3\\) be the oriented double cover branched over \\(C\\), and give \\(K'=\\pi^{-1}(K)\\) and \\(C'=\\pi^{-1}(C)\\) the lifted orientations.\nThen \\(K'\\) is a knot and\n\\[\n\\Delta_{K'\\cup C'}(t,y^2)\\doteq\\Delta_{K\\cup C}(t,y)\\Delta_{K\\cup C}(t,-y).\n\\]\nThe variables \\(t,y\\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \\(\\doteq\\) means equality up to multiplication by \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
    "starting_statement": "Let \\(K\\subset S^3\\) be a smooth oriented knot admitting an orientation-reversing diffeomorphism \\(h:S^3\\to S^3\\) with \\(h^2=\\operatorname{id}\\), \\(h(K)=K\\), and \\(h|_K\\) preserving orientation.\nThere exists \\(\\varphi\\in\\mathbb Z[w]\\) such that\n\\[\n\\nabla_K(z)=\\varphi(z^2)^2.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\).",
    "target_statement": "Let \\(L=K\\cup C\\subset S^3\\) be a smooth oriented two-component link with \\(C\\) unknotted and \\(\\operatorname{lk}(K,C)\\) odd. Let \\(\\pi:S^3\\to S^3\\) be the oriented double cover branched over \\(C\\), and give \\(K'=\\pi^{-1}(K)\\) and \\(C'=\\pi^{-1}(C)\\) the lifted orientations.\nThen \\(K'\\) is a knot and\n\\[\n\\Delta_{K'\\cup C'}(t,y^2)\\doteq\\Delta_{K\\cup C}(t,y)\\Delta_{K\\cup C}(t,-y).\n\\]\nThe variables \\(t,y\\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \\(\\doteq\\) means equality up to multiplication by \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
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  "KT-1-H2-to-H4": {
    "id": "KT-1-H2-to-H4",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(K\\subset S^3\\) be a smooth oriented knot admitting an orientation-reversing diffeomorphism \\(h:S^3\\to S^3\\) with \\(h^2=\\operatorname{id}\\), \\(h(K)=K\\), and \\(h|_K\\) preserving orientation.\nThere exists \\(\\varphi\\in\\mathbb Z[w]\\) such that\n\\[\n\\nabla_K(z)=\\varphi(z^2)^2.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\).\n\nProve H4: Identify \\(S^3\\) with \\(\\{(z_1,z_2)\\in\\mathbb C^2:|z_1|^2+|z_2|^2=1\\}\\). For integers \\(a\\ge2\\) and odd \\(c\\), put\n\\[\n\\rho(z_1,z_2)=(e^{2\\pi ic/2^a}z_1,\\overline{z_2}),\\qquad C=\\{z_1=0\\}.\n\\]\nLet \\(K\\subset S^3\\) be a smooth oriented knot disjoint from \\(C\\), with \\(\\rho(K)=K\\), \\(\\rho|_K\\) preserving orientation, and \\(\\operatorname{lk}(K,C)\\) odd.\nThere exists \\(Q\\in\\mathbb Z[U,V]\\) such that\n\\[\n\\Delta_{K\\cup C}(t,y)\\doteq Q(t+t^{-1},y+y^{-1}).\n\\]\nThe variables \\(t,y\\) correspond to \\(K,C\\), respectively, and \\(\\doteq\\) denotes equality up to a Laurent unit \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
    "starting_statement": "Let \\(K\\subset S^3\\) be a smooth oriented knot admitting an orientation-reversing diffeomorphism \\(h:S^3\\to S^3\\) with \\(h^2=\\operatorname{id}\\), \\(h(K)=K\\), and \\(h|_K\\) preserving orientation.\nThere exists \\(\\varphi\\in\\mathbb Z[w]\\) such that\n\\[\n\\nabla_K(z)=\\varphi(z^2)^2.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\).",
    "target_statement": "Identify \\(S^3\\) with \\(\\{(z_1,z_2)\\in\\mathbb C^2:|z_1|^2+|z_2|^2=1\\}\\). For integers \\(a\\ge2\\) and odd \\(c\\), put\n\\[\n\\rho(z_1,z_2)=(e^{2\\pi ic/2^a}z_1,\\overline{z_2}),\\qquad C=\\{z_1=0\\}.\n\\]\nLet \\(K\\subset S^3\\) be a smooth oriented knot disjoint from \\(C\\), with \\(\\rho(K)=K\\), \\(\\rho|_K\\) preserving orientation, and \\(\\operatorname{lk}(K,C)\\) odd.\nThere exists \\(Q\\in\\mathbb Z[U,V]\\) such that\n\\[\n\\Delta_{K\\cup C}(t,y)\\doteq Q(t+t^{-1},y+y^{-1}).\n\\]\nThe variables \\(t,y\\) correspond to \\(K,C\\), respectively, and \\(\\doteq\\) denotes equality up to a Laurent unit \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
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  "KT-1-H2-to-H5": {
    "id": "KT-1-H2-to-H5",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(K\\subset S^3\\) be a smooth oriented knot admitting an orientation-reversing diffeomorphism \\(h:S^3\\to S^3\\) with \\(h^2=\\operatorname{id}\\), \\(h(K)=K\\), and \\(h|_K\\) preserving orientation.\nThere exists \\(\\varphi\\in\\mathbb Z[w]\\) such that\n\\[\n\\nabla_K(z)=\\varphi(z^2)^2.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\).\n\nProve H5: Let \\(K\\subset S^3\\) be an oriented knot admitting a finite-order orientation-reversing diffeomorphism of \\(S^3\\) that preserves the orientation of \\(K\\). Let \\(\\nabla_K(z)\\in\\mathbb Z[z]\\) be its Conway polynomial, normalized by \\(\\nabla_{\\rm unknot}=1\\).\nThere exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4,\n\\]\nwhere the congruence is coefficientwise.",
    "starting_statement": "Let \\(K\\subset S^3\\) be a smooth oriented knot admitting an orientation-reversing diffeomorphism \\(h:S^3\\to S^3\\) with \\(h^2=\\operatorname{id}\\), \\(h(K)=K\\), and \\(h|_K\\) preserving orientation.\nThere exists \\(\\varphi\\in\\mathbb Z[w]\\) such that\n\\[\n\\nabla_K(z)=\\varphi(z^2)^2.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\).",
    "target_statement": "Let \\(K\\subset S^3\\) be an oriented knot admitting a finite-order orientation-reversing diffeomorphism of \\(S^3\\) that preserves the orientation of \\(K\\). Let \\(\\nabla_K(z)\\in\\mathbb Z[z]\\) be its Conway polynomial, normalized by \\(\\nabla_{\\rm unknot}=1\\).\nThere exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4,\n\\]\nwhere the congruence is coefficientwise.",
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  "KT-1-H2-to-H6": {
    "id": "KT-1-H2-to-H6",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(K\\subset S^3\\) be a smooth oriented knot admitting an orientation-reversing diffeomorphism \\(h:S^3\\to S^3\\) with \\(h^2=\\operatorname{id}\\), \\(h(K)=K\\), and \\(h|_K\\) preserving orientation.\nThere exists \\(\\varphi\\in\\mathbb Z[w]\\) such that\n\\[\n\\nabla_K(z)=\\varphi(z^2)^2.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\).\n\nProve H6: A knot \\(K\\subset S^3\\) is called amphicheiral if an orientation-reversing homeomorphism of \\(S^3\\) maps \\(K\\) to itself; the homeomorphism may preserve or reverse the knot orientation.\nFor every amphicheiral knot \\(K\\), including the unknot, there exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\), and the congruence is coefficientwise.",
    "starting_statement": "Let \\(K\\subset S^3\\) be a smooth oriented knot admitting an orientation-reversing diffeomorphism \\(h:S^3\\to S^3\\) with \\(h^2=\\operatorname{id}\\), \\(h(K)=K\\), and \\(h|_K\\) preserving orientation.\nThere exists \\(\\varphi\\in\\mathbb Z[w]\\) such that\n\\[\n\\nabla_K(z)=\\varphi(z^2)^2.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\).",
    "target_statement": "A knot \\(K\\subset S^3\\) is called amphicheiral if an orientation-reversing homeomorphism of \\(S^3\\) maps \\(K\\) to itself; the homeomorphism may preserve or reverse the knot orientation.\nFor every amphicheiral knot \\(K\\), including the unknot, there exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\), and the congruence is coefficientwise.",
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  "KT-1-H3-to-H4": {
    "id": "KT-1-H3-to-H4",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H3: Let \\(L=K\\cup C\\subset S^3\\) be a smooth oriented two-component link with \\(C\\) unknotted and \\(\\operatorname{lk}(K,C)\\) odd. Let \\(\\pi:S^3\\to S^3\\) be the oriented double cover branched over \\(C\\), and give \\(K'=\\pi^{-1}(K)\\) and \\(C'=\\pi^{-1}(C)\\) the lifted orientations.\nThen \\(K'\\) is a knot and\n\\[\n\\Delta_{K'\\cup C'}(t,y^2)\\doteq\\Delta_{K\\cup C}(t,y)\\Delta_{K\\cup C}(t,-y).\n\\]\nThe variables \\(t,y\\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \\(\\doteq\\) means equality up to multiplication by \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).\n\nProve H4: Identify \\(S^3\\) with \\(\\{(z_1,z_2)\\in\\mathbb C^2:|z_1|^2+|z_2|^2=1\\}\\). For integers \\(a\\ge2\\) and odd \\(c\\), put\n\\[\n\\rho(z_1,z_2)=(e^{2\\pi ic/2^a}z_1,\\overline{z_2}),\\qquad C=\\{z_1=0\\}.\n\\]\nLet \\(K\\subset S^3\\) be a smooth oriented knot disjoint from \\(C\\), with \\(\\rho(K)=K\\), \\(\\rho|_K\\) preserving orientation, and \\(\\operatorname{lk}(K,C)\\) odd.\nThere exists \\(Q\\in\\mathbb Z[U,V]\\) such that\n\\[\n\\Delta_{K\\cup C}(t,y)\\doteq Q(t+t^{-1},y+y^{-1}).\n\\]\nThe variables \\(t,y\\) correspond to \\(K,C\\), respectively, and \\(\\doteq\\) denotes equality up to a Laurent unit \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
    "starting_statement": "Let \\(L=K\\cup C\\subset S^3\\) be a smooth oriented two-component link with \\(C\\) unknotted and \\(\\operatorname{lk}(K,C)\\) odd. Let \\(\\pi:S^3\\to S^3\\) be the oriented double cover branched over \\(C\\), and give \\(K'=\\pi^{-1}(K)\\) and \\(C'=\\pi^{-1}(C)\\) the lifted orientations.\nThen \\(K'\\) is a knot and\n\\[\n\\Delta_{K'\\cup C'}(t,y^2)\\doteq\\Delta_{K\\cup C}(t,y)\\Delta_{K\\cup C}(t,-y).\n\\]\nThe variables \\(t,y\\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \\(\\doteq\\) means equality up to multiplication by \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
    "target_statement": "Identify \\(S^3\\) with \\(\\{(z_1,z_2)\\in\\mathbb C^2:|z_1|^2+|z_2|^2=1\\}\\). For integers \\(a\\ge2\\) and odd \\(c\\), put\n\\[\n\\rho(z_1,z_2)=(e^{2\\pi ic/2^a}z_1,\\overline{z_2}),\\qquad C=\\{z_1=0\\}.\n\\]\nLet \\(K\\subset S^3\\) be a smooth oriented knot disjoint from \\(C\\), with \\(\\rho(K)=K\\), \\(\\rho|_K\\) preserving orientation, and \\(\\operatorname{lk}(K,C)\\) odd.\nThere exists \\(Q\\in\\mathbb Z[U,V]\\) such that\n\\[\n\\Delta_{K\\cup C}(t,y)\\doteq Q(t+t^{-1},y+y^{-1}).\n\\]\nThe variables \\(t,y\\) correspond to \\(K,C\\), respectively, and \\(\\doteq\\) denotes equality up to a Laurent unit \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
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  "KT-1-H3-to-H5": {
    "id": "KT-1-H3-to-H5",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H3: Let \\(L=K\\cup C\\subset S^3\\) be a smooth oriented two-component link with \\(C\\) unknotted and \\(\\operatorname{lk}(K,C)\\) odd. Let \\(\\pi:S^3\\to S^3\\) be the oriented double cover branched over \\(C\\), and give \\(K'=\\pi^{-1}(K)\\) and \\(C'=\\pi^{-1}(C)\\) the lifted orientations.\nThen \\(K'\\) is a knot and\n\\[\n\\Delta_{K'\\cup C'}(t,y^2)\\doteq\\Delta_{K\\cup C}(t,y)\\Delta_{K\\cup C}(t,-y).\n\\]\nThe variables \\(t,y\\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \\(\\doteq\\) means equality up to multiplication by \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).\n\nProve H5: Let \\(K\\subset S^3\\) be an oriented knot admitting a finite-order orientation-reversing diffeomorphism of \\(S^3\\) that preserves the orientation of \\(K\\). Let \\(\\nabla_K(z)\\in\\mathbb Z[z]\\) be its Conway polynomial, normalized by \\(\\nabla_{\\rm unknot}=1\\).\nThere exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4,\n\\]\nwhere the congruence is coefficientwise.",
    "starting_statement": "Let \\(L=K\\cup C\\subset S^3\\) be a smooth oriented two-component link with \\(C\\) unknotted and \\(\\operatorname{lk}(K,C)\\) odd. Let \\(\\pi:S^3\\to S^3\\) be the oriented double cover branched over \\(C\\), and give \\(K'=\\pi^{-1}(K)\\) and \\(C'=\\pi^{-1}(C)\\) the lifted orientations.\nThen \\(K'\\) is a knot and\n\\[\n\\Delta_{K'\\cup C'}(t,y^2)\\doteq\\Delta_{K\\cup C}(t,y)\\Delta_{K\\cup C}(t,-y).\n\\]\nThe variables \\(t,y\\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \\(\\doteq\\) means equality up to multiplication by \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
    "target_statement": "Let \\(K\\subset S^3\\) be an oriented knot admitting a finite-order orientation-reversing diffeomorphism of \\(S^3\\) that preserves the orientation of \\(K\\). Let \\(\\nabla_K(z)\\in\\mathbb Z[z]\\) be its Conway polynomial, normalized by \\(\\nabla_{\\rm unknot}=1\\).\nThere exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4,\n\\]\nwhere the congruence is coefficientwise.",
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  "KT-1-H3-to-H6": {
    "id": "KT-1-H3-to-H6",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H3: Let \\(L=K\\cup C\\subset S^3\\) be a smooth oriented two-component link with \\(C\\) unknotted and \\(\\operatorname{lk}(K,C)\\) odd. Let \\(\\pi:S^3\\to S^3\\) be the oriented double cover branched over \\(C\\), and give \\(K'=\\pi^{-1}(K)\\) and \\(C'=\\pi^{-1}(C)\\) the lifted orientations.\nThen \\(K'\\) is a knot and\n\\[\n\\Delta_{K'\\cup C'}(t,y^2)\\doteq\\Delta_{K\\cup C}(t,y)\\Delta_{K\\cup C}(t,-y).\n\\]\nThe variables \\(t,y\\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \\(\\doteq\\) means equality up to multiplication by \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).\n\nProve H6: A knot \\(K\\subset S^3\\) is called amphicheiral if an orientation-reversing homeomorphism of \\(S^3\\) maps \\(K\\) to itself; the homeomorphism may preserve or reverse the knot orientation.\nFor every amphicheiral knot \\(K\\), including the unknot, there exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\), and the congruence is coefficientwise.",
    "starting_statement": "Let \\(L=K\\cup C\\subset S^3\\) be a smooth oriented two-component link with \\(C\\) unknotted and \\(\\operatorname{lk}(K,C)\\) odd. Let \\(\\pi:S^3\\to S^3\\) be the oriented double cover branched over \\(C\\), and give \\(K'=\\pi^{-1}(K)\\) and \\(C'=\\pi^{-1}(C)\\) the lifted orientations.\nThen \\(K'\\) is a knot and\n\\[\n\\Delta_{K'\\cup C'}(t,y^2)\\doteq\\Delta_{K\\cup C}(t,y)\\Delta_{K\\cup C}(t,-y).\n\\]\nThe variables \\(t,y\\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \\(\\doteq\\) means equality up to multiplication by \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
    "target_statement": "A knot \\(K\\subset S^3\\) is called amphicheiral if an orientation-reversing homeomorphism of \\(S^3\\) maps \\(K\\) to itself; the homeomorphism may preserve or reverse the knot orientation.\nFor every amphicheiral knot \\(K\\), including the unknot, there exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\), and the congruence is coefficientwise.",
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  "KT-1-H4-to-H5": {
    "id": "KT-1-H4-to-H5",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 4,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H4: Identify \\(S^3\\) with \\(\\{(z_1,z_2)\\in\\mathbb C^2:|z_1|^2+|z_2|^2=1\\}\\). For integers \\(a\\ge2\\) and odd \\(c\\), put\n\\[\n\\rho(z_1,z_2)=(e^{2\\pi ic/2^a}z_1,\\overline{z_2}),\\qquad C=\\{z_1=0\\}.\n\\]\nLet \\(K\\subset S^3\\) be a smooth oriented knot disjoint from \\(C\\), with \\(\\rho(K)=K\\), \\(\\rho|_K\\) preserving orientation, and \\(\\operatorname{lk}(K,C)\\) odd.\nThere exists \\(Q\\in\\mathbb Z[U,V]\\) such that\n\\[\n\\Delta_{K\\cup C}(t,y)\\doteq Q(t+t^{-1},y+y^{-1}).\n\\]\nThe variables \\(t,y\\) correspond to \\(K,C\\), respectively, and \\(\\doteq\\) denotes equality up to a Laurent unit \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).\n\nProve H5: Let \\(K\\subset S^3\\) be an oriented knot admitting a finite-order orientation-reversing diffeomorphism of \\(S^3\\) that preserves the orientation of \\(K\\). Let \\(\\nabla_K(z)\\in\\mathbb Z[z]\\) be its Conway polynomial, normalized by \\(\\nabla_{\\rm unknot}=1\\).\nThere exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4,\n\\]\nwhere the congruence is coefficientwise.",
    "starting_statement": "Identify \\(S^3\\) with \\(\\{(z_1,z_2)\\in\\mathbb C^2:|z_1|^2+|z_2|^2=1\\}\\). For integers \\(a\\ge2\\) and odd \\(c\\), put\n\\[\n\\rho(z_1,z_2)=(e^{2\\pi ic/2^a}z_1,\\overline{z_2}),\\qquad C=\\{z_1=0\\}.\n\\]\nLet \\(K\\subset S^3\\) be a smooth oriented knot disjoint from \\(C\\), with \\(\\rho(K)=K\\), \\(\\rho|_K\\) preserving orientation, and \\(\\operatorname{lk}(K,C)\\) odd.\nThere exists \\(Q\\in\\mathbb Z[U,V]\\) such that\n\\[\n\\Delta_{K\\cup C}(t,y)\\doteq Q(t+t^{-1},y+y^{-1}).\n\\]\nThe variables \\(t,y\\) correspond to \\(K,C\\), respectively, and \\(\\doteq\\) denotes equality up to a Laurent unit \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
    "target_statement": "Let \\(K\\subset S^3\\) be an oriented knot admitting a finite-order orientation-reversing diffeomorphism of \\(S^3\\) that preserves the orientation of \\(K\\). Let \\(\\nabla_K(z)\\in\\mathbb Z[z]\\) be its Conway polynomial, normalized by \\(\\nabla_{\\rm unknot}=1\\).\nThere exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4,\n\\]\nwhere the congruence is coefficientwise.",
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  "KT-1-H4-to-H6": {
    "id": "KT-1-H4-to-H6",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 4,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H4: Identify \\(S^3\\) with \\(\\{(z_1,z_2)\\in\\mathbb C^2:|z_1|^2+|z_2|^2=1\\}\\). For integers \\(a\\ge2\\) and odd \\(c\\), put\n\\[\n\\rho(z_1,z_2)=(e^{2\\pi ic/2^a}z_1,\\overline{z_2}),\\qquad C=\\{z_1=0\\}.\n\\]\nLet \\(K\\subset S^3\\) be a smooth oriented knot disjoint from \\(C\\), with \\(\\rho(K)=K\\), \\(\\rho|_K\\) preserving orientation, and \\(\\operatorname{lk}(K,C)\\) odd.\nThere exists \\(Q\\in\\mathbb Z[U,V]\\) such that\n\\[\n\\Delta_{K\\cup C}(t,y)\\doteq Q(t+t^{-1},y+y^{-1}).\n\\]\nThe variables \\(t,y\\) correspond to \\(K,C\\), respectively, and \\(\\doteq\\) denotes equality up to a Laurent unit \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).\n\nProve H6: A knot \\(K\\subset S^3\\) is called amphicheiral if an orientation-reversing homeomorphism of \\(S^3\\) maps \\(K\\) to itself; the homeomorphism may preserve or reverse the knot orientation.\nFor every amphicheiral knot \\(K\\), including the unknot, there exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\), and the congruence is coefficientwise.",
    "starting_statement": "Identify \\(S^3\\) with \\(\\{(z_1,z_2)\\in\\mathbb C^2:|z_1|^2+|z_2|^2=1\\}\\). For integers \\(a\\ge2\\) and odd \\(c\\), put\n\\[\n\\rho(z_1,z_2)=(e^{2\\pi ic/2^a}z_1,\\overline{z_2}),\\qquad C=\\{z_1=0\\}.\n\\]\nLet \\(K\\subset S^3\\) be a smooth oriented knot disjoint from \\(C\\), with \\(\\rho(K)=K\\), \\(\\rho|_K\\) preserving orientation, and \\(\\operatorname{lk}(K,C)\\) odd.\nThere exists \\(Q\\in\\mathbb Z[U,V]\\) such that\n\\[\n\\Delta_{K\\cup C}(t,y)\\doteq Q(t+t^{-1},y+y^{-1}).\n\\]\nThe variables \\(t,y\\) correspond to \\(K,C\\), respectively, and \\(\\doteq\\) denotes equality up to a Laurent unit \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
    "target_statement": "A knot \\(K\\subset S^3\\) is called amphicheiral if an orientation-reversing homeomorphism of \\(S^3\\) maps \\(K\\) to itself; the homeomorphism may preserve or reverse the knot orientation.\nFor every amphicheiral knot \\(K\\), including the unknot, there exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\), and the congruence is coefficientwise.",
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  "KT-1-H5-to-H6": {
    "id": "KT-1-H5-to-H6",
    "problem_id": "KT-1",
    "problem_name": "Amphicheiral knot polynomials",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 5,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H5: Let \\(K\\subset S^3\\) be an oriented knot admitting a finite-order orientation-reversing diffeomorphism of \\(S^3\\) that preserves the orientation of \\(K\\). Let \\(\\nabla_K(z)\\in\\mathbb Z[z]\\) be its Conway polynomial, normalized by \\(\\nabla_{\\rm unknot}=1\\).\nThere exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4,\n\\]\nwhere the congruence is coefficientwise.\n\nProve H6: A knot \\(K\\subset S^3\\) is called amphicheiral if an orientation-reversing homeomorphism of \\(S^3\\) maps \\(K\\) to itself; the homeomorphism may preserve or reverse the knot orientation.\nFor every amphicheiral knot \\(K\\), including the unknot, there exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\), and the congruence is coefficientwise.",
    "starting_statement": "Let \\(K\\subset S^3\\) be an oriented knot admitting a finite-order orientation-reversing diffeomorphism of \\(S^3\\) that preserves the orientation of \\(K\\). Let \\(\\nabla_K(z)\\in\\mathbb Z[z]\\) be its Conway polynomial, normalized by \\(\\nabla_{\\rm unknot}=1\\).\nThere exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4,\n\\]\nwhere the congruence is coefficientwise.",
    "target_statement": "A knot \\(K\\subset S^3\\) is called amphicheiral if an orientation-reversing homeomorphism of \\(S^3\\) maps \\(K\\) to itself; the homeomorphism may preserve or reverse the knot orientation.\nFor every amphicheiral knot \\(K\\), including the unknot, there exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\), and the congruence is coefficientwise.",
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  "NE-1-H0-to-H1": {
    "id": "NE-1-H0-to-H1",
    "problem_id": "NE-1",
    "problem_name": "Periodic BBM",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 1,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H1: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFor every \\(x\\in\\omega\\) and \\(t\\in\\mathbb R\\),\n\\[\nz(x,t)=w(x,t)=\\partial_xw(x,t)=0.\n\\]",
    "starting_statement": null,
    "target_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFor every \\(x\\in\\omega\\) and \\(t\\in\\mathbb R\\),\n\\[\nz(x,t)=w(x,t)=\\partial_xw(x,t)=0.\n\\]",
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  "NE-1-H0-to-H2": {
    "id": "NE-1-H0-to-H2",
    "problem_id": "NE-1",
    "problem_name": "Periodic BBM",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 2,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H2: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFix \\(x_0\\in\\omega\\) and use the interval coordinate \\(x\\in[-\\pi,\\pi]\\) on \\(\\mathbb T\\) with both endpoints representing \\(x_0\\). For \\(\\alpha>2\\), define\n\\[\nH_\\beta(s)=\\beta s^2/2+s^3/6,\\quad\nN_\\alpha(t)=\\int_{-\\pi}^{\\pi}e^{\\alpha x}H_\\beta(z(x,t))\\,dx,\\quad\nW_\\alpha(x,t)=e^{\\alpha x/2}w(x,t).\n\\]\nThen \\(N_\\alpha\\in C^1(\\mathbb R)\\), and for every \\(t\\in\\mathbb R\\),\n\\[\nN_\\alpha'(t)=-\\frac\\alpha2\\int_{-\\pi}^{\\pi}\n\\left[(\\partial_xW_\\alpha)^2+(\\alpha^2/4-1)W_\\alpha^2\\right]dx.\n\\]",
    "starting_statement": null,
    "target_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFix \\(x_0\\in\\omega\\) and use the interval coordinate \\(x\\in[-\\pi,\\pi]\\) on \\(\\mathbb T\\) with both endpoints representing \\(x_0\\). For \\(\\alpha>2\\), define\n\\[\nH_\\beta(s)=\\beta s^2/2+s^3/6,\\quad\nN_\\alpha(t)=\\int_{-\\pi}^{\\pi}e^{\\alpha x}H_\\beta(z(x,t))\\,dx,\\quad\nW_\\alpha(x,t)=e^{\\alpha x/2}w(x,t).\n\\]\nThen \\(N_\\alpha\\in C^1(\\mathbb R)\\), and for every \\(t\\in\\mathbb R\\),\n\\[\nN_\\alpha'(t)=-\\frac\\alpha2\\int_{-\\pi}^{\\pi}\n\\left[(\\partial_xW_\\alpha)^2+(\\alpha^2/4-1)W_\\alpha^2\\right]dx.\n\\]",
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  "NE-1-H0-to-H3": {
    "id": "NE-1-H0-to-H3",
    "problem_id": "NE-1",
    "problem_name": "Periodic BBM",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H3: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nThe integral over the entire time axis is finite:\n\\[\n\\int_{\\mathbb R}\\|w(t)\\|_{L^2(\\mathbb T)}^2\\,dt<\\infty.\n\\]",
    "starting_statement": null,
    "target_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nThe integral over the entire time axis is finite:\n\\[\n\\int_{\\mathbb R}\\|w(t)\\|_{L^2(\\mathbb T)}^2\\,dt<\\infty.\n\\]",
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  "NE-1-H0-to-H4": {
    "id": "NE-1-H0-to-H4",
    "problem_id": "NE-1",
    "problem_name": "Periodic BBM",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H4: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\). Suppose \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfies\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. If there exist \\(C\\in\\mathbb R\\), \\(T_0>0\\), and a nonempty open set \\(\\omega\\subset\\mathbb T\\) such that \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), then\n\\[\nu(x,t)=C\\qquad\\text{for every }(x,t)\\in\\mathbb T\\times\\mathbb R.\n\\]",
    "starting_statement": null,
    "target_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\). Suppose \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfies\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. If there exist \\(C\\in\\mathbb R\\), \\(T_0>0\\), and a nonempty open set \\(\\omega\\subset\\mathbb T\\) such that \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), then\n\\[\nu(x,t)=C\\qquad\\text{for every }(x,t)\\in\\mathbb T\\times\\mathbb R.\n\\]",
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  "NE-1-H1-to-H2": {
    "id": "NE-1-H1-to-H2",
    "problem_id": "NE-1",
    "problem_name": "Periodic BBM",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 2,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFor every \\(x\\in\\omega\\) and \\(t\\in\\mathbb R\\),\n\\[\nz(x,t)=w(x,t)=\\partial_xw(x,t)=0.\n\\]\n\nProve H2: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFix \\(x_0\\in\\omega\\) and use the interval coordinate \\(x\\in[-\\pi,\\pi]\\) on \\(\\mathbb T\\) with both endpoints representing \\(x_0\\). For \\(\\alpha>2\\), define\n\\[\nH_\\beta(s)=\\beta s^2/2+s^3/6,\\quad\nN_\\alpha(t)=\\int_{-\\pi}^{\\pi}e^{\\alpha x}H_\\beta(z(x,t))\\,dx,\\quad\nW_\\alpha(x,t)=e^{\\alpha x/2}w(x,t).\n\\]\nThen \\(N_\\alpha\\in C^1(\\mathbb R)\\), and for every \\(t\\in\\mathbb R\\),\n\\[\nN_\\alpha'(t)=-\\frac\\alpha2\\int_{-\\pi}^{\\pi}\n\\left[(\\partial_xW_\\alpha)^2+(\\alpha^2/4-1)W_\\alpha^2\\right]dx.\n\\]",
    "starting_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFor every \\(x\\in\\omega\\) and \\(t\\in\\mathbb R\\),\n\\[\nz(x,t)=w(x,t)=\\partial_xw(x,t)=0.\n\\]",
    "target_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFix \\(x_0\\in\\omega\\) and use the interval coordinate \\(x\\in[-\\pi,\\pi]\\) on \\(\\mathbb T\\) with both endpoints representing \\(x_0\\). For \\(\\alpha>2\\), define\n\\[\nH_\\beta(s)=\\beta s^2/2+s^3/6,\\quad\nN_\\alpha(t)=\\int_{-\\pi}^{\\pi}e^{\\alpha x}H_\\beta(z(x,t))\\,dx,\\quad\nW_\\alpha(x,t)=e^{\\alpha x/2}w(x,t).\n\\]\nThen \\(N_\\alpha\\in C^1(\\mathbb R)\\), and for every \\(t\\in\\mathbb R\\),\n\\[\nN_\\alpha'(t)=-\\frac\\alpha2\\int_{-\\pi}^{\\pi}\n\\left[(\\partial_xW_\\alpha)^2+(\\alpha^2/4-1)W_\\alpha^2\\right]dx.\n\\]",
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  "NE-1-H1-to-H3": {
    "id": "NE-1-H1-to-H3",
    "problem_id": "NE-1",
    "problem_name": "Periodic BBM",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFor every \\(x\\in\\omega\\) and \\(t\\in\\mathbb R\\),\n\\[\nz(x,t)=w(x,t)=\\partial_xw(x,t)=0.\n\\]\n\nProve H3: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nThe integral over the entire time axis is finite:\n\\[\n\\int_{\\mathbb R}\\|w(t)\\|_{L^2(\\mathbb T)}^2\\,dt<\\infty.\n\\]",
    "starting_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFor every \\(x\\in\\omega\\) and \\(t\\in\\mathbb R\\),\n\\[\nz(x,t)=w(x,t)=\\partial_xw(x,t)=0.\n\\]",
    "target_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nThe integral over the entire time axis is finite:\n\\[\n\\int_{\\mathbb R}\\|w(t)\\|_{L^2(\\mathbb T)}^2\\,dt<\\infty.\n\\]",
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  "NE-1-H1-to-H4": {
    "id": "NE-1-H1-to-H4",
    "problem_id": "NE-1",
    "problem_name": "Periodic BBM",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFor every \\(x\\in\\omega\\) and \\(t\\in\\mathbb R\\),\n\\[\nz(x,t)=w(x,t)=\\partial_xw(x,t)=0.\n\\]\n\nProve H4: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\). Suppose \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfies\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. If there exist \\(C\\in\\mathbb R\\), \\(T_0>0\\), and a nonempty open set \\(\\omega\\subset\\mathbb T\\) such that \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), then\n\\[\nu(x,t)=C\\qquad\\text{for every }(x,t)\\in\\mathbb T\\times\\mathbb R.\n\\]",
    "starting_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFor every \\(x\\in\\omega\\) and \\(t\\in\\mathbb R\\),\n\\[\nz(x,t)=w(x,t)=\\partial_xw(x,t)=0.\n\\]",
    "target_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\). Suppose \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfies\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. If there exist \\(C\\in\\mathbb R\\), \\(T_0>0\\), and a nonempty open set \\(\\omega\\subset\\mathbb T\\) such that \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), then\n\\[\nu(x,t)=C\\qquad\\text{for every }(x,t)\\in\\mathbb T\\times\\mathbb R.\n\\]",
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  "NE-1-H2-to-H3": {
    "id": "NE-1-H2-to-H3",
    "problem_id": "NE-1",
    "problem_name": "Periodic BBM",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFix \\(x_0\\in\\omega\\) and use the interval coordinate \\(x\\in[-\\pi,\\pi]\\) on \\(\\mathbb T\\) with both endpoints representing \\(x_0\\). For \\(\\alpha>2\\), define\n\\[\nH_\\beta(s)=\\beta s^2/2+s^3/6,\\quad\nN_\\alpha(t)=\\int_{-\\pi}^{\\pi}e^{\\alpha x}H_\\beta(z(x,t))\\,dx,\\quad\nW_\\alpha(x,t)=e^{\\alpha x/2}w(x,t).\n\\]\nThen \\(N_\\alpha\\in C^1(\\mathbb R)\\), and for every \\(t\\in\\mathbb R\\),\n\\[\nN_\\alpha'(t)=-\\frac\\alpha2\\int_{-\\pi}^{\\pi}\n\\left[(\\partial_xW_\\alpha)^2+(\\alpha^2/4-1)W_\\alpha^2\\right]dx.\n\\]\n\nProve H3: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nThe integral over the entire time axis is finite:\n\\[\n\\int_{\\mathbb R}\\|w(t)\\|_{L^2(\\mathbb T)}^2\\,dt<\\infty.\n\\]",
    "starting_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFix \\(x_0\\in\\omega\\) and use the interval coordinate \\(x\\in[-\\pi,\\pi]\\) on \\(\\mathbb T\\) with both endpoints representing \\(x_0\\). For \\(\\alpha>2\\), define\n\\[\nH_\\beta(s)=\\beta s^2/2+s^3/6,\\quad\nN_\\alpha(t)=\\int_{-\\pi}^{\\pi}e^{\\alpha x}H_\\beta(z(x,t))\\,dx,\\quad\nW_\\alpha(x,t)=e^{\\alpha x/2}w(x,t).\n\\]\nThen \\(N_\\alpha\\in C^1(\\mathbb R)\\), and for every \\(t\\in\\mathbb R\\),\n\\[\nN_\\alpha'(t)=-\\frac\\alpha2\\int_{-\\pi}^{\\pi}\n\\left[(\\partial_xW_\\alpha)^2+(\\alpha^2/4-1)W_\\alpha^2\\right]dx.\n\\]",
    "target_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nThe integral over the entire time axis is finite:\n\\[\n\\int_{\\mathbb R}\\|w(t)\\|_{L^2(\\mathbb T)}^2\\,dt<\\infty.\n\\]",
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  "NE-1-H2-to-H4": {
    "id": "NE-1-H2-to-H4",
    "problem_id": "NE-1",
    "problem_name": "Periodic BBM",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFix \\(x_0\\in\\omega\\) and use the interval coordinate \\(x\\in[-\\pi,\\pi]\\) on \\(\\mathbb T\\) with both endpoints representing \\(x_0\\). For \\(\\alpha>2\\), define\n\\[\nH_\\beta(s)=\\beta s^2/2+s^3/6,\\quad\nN_\\alpha(t)=\\int_{-\\pi}^{\\pi}e^{\\alpha x}H_\\beta(z(x,t))\\,dx,\\quad\nW_\\alpha(x,t)=e^{\\alpha x/2}w(x,t).\n\\]\nThen \\(N_\\alpha\\in C^1(\\mathbb R)\\), and for every \\(t\\in\\mathbb R\\),\n\\[\nN_\\alpha'(t)=-\\frac\\alpha2\\int_{-\\pi}^{\\pi}\n\\left[(\\partial_xW_\\alpha)^2+(\\alpha^2/4-1)W_\\alpha^2\\right]dx.\n\\]\n\nProve H4: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\). Suppose \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfies\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. If there exist \\(C\\in\\mathbb R\\), \\(T_0>0\\), and a nonempty open set \\(\\omega\\subset\\mathbb T\\) such that \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), then\n\\[\nu(x,t)=C\\qquad\\text{for every }(x,t)\\in\\mathbb T\\times\\mathbb R.\n\\]",
    "starting_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFix \\(x_0\\in\\omega\\) and use the interval coordinate \\(x\\in[-\\pi,\\pi]\\) on \\(\\mathbb T\\) with both endpoints representing \\(x_0\\). For \\(\\alpha>2\\), define\n\\[\nH_\\beta(s)=\\beta s^2/2+s^3/6,\\quad\nN_\\alpha(t)=\\int_{-\\pi}^{\\pi}e^{\\alpha x}H_\\beta(z(x,t))\\,dx,\\quad\nW_\\alpha(x,t)=e^{\\alpha x/2}w(x,t).\n\\]\nThen \\(N_\\alpha\\in C^1(\\mathbb R)\\), and for every \\(t\\in\\mathbb R\\),\n\\[\nN_\\alpha'(t)=-\\frac\\alpha2\\int_{-\\pi}^{\\pi}\n\\left[(\\partial_xW_\\alpha)^2+(\\alpha^2/4-1)W_\\alpha^2\\right]dx.\n\\]",
    "target_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\). Suppose \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfies\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. If there exist \\(C\\in\\mathbb R\\), \\(T_0>0\\), and a nonempty open set \\(\\omega\\subset\\mathbb T\\) such that \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), then\n\\[\nu(x,t)=C\\qquad\\text{for every }(x,t)\\in\\mathbb T\\times\\mathbb R.\n\\]",
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  "NE-1-H3-to-H4": {
    "id": "NE-1-H3-to-H4",
    "problem_id": "NE-1",
    "problem_name": "Periodic BBM",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H3: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nThe integral over the entire time axis is finite:\n\\[\n\\int_{\\mathbb R}\\|w(t)\\|_{L^2(\\mathbb T)}^2\\,dt<\\infty.\n\\]\n\nProve H4: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\). Suppose \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfies\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. If there exist \\(C\\in\\mathbb R\\), \\(T_0>0\\), and a nonempty open set \\(\\omega\\subset\\mathbb T\\) such that \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), then\n\\[\nu(x,t)=C\\qquad\\text{for every }(x,t)\\in\\mathbb T\\times\\mathbb R.\n\\]",
    "starting_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nThe integral over the entire time axis is finite:\n\\[\n\\int_{\\mathbb R}\\|w(t)\\|_{L^2(\\mathbb T)}^2\\,dt<\\infty.\n\\]",
    "target_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\). Suppose \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfies\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. If there exist \\(C\\in\\mathbb R\\), \\(T_0>0\\), and a nonempty open set \\(\\omega\\subset\\mathbb T\\) such that \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), then\n\\[\nu(x,t)=C\\qquad\\text{for every }(x,t)\\in\\mathbb T\\times\\mathbb R.\n\\]",
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  "OA-1-H0-to-H1": {
    "id": "OA-1-H0-to-H1",
    "problem_id": "OA-1",
    "problem_name": "Scalar Crouzeix inequality",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 1,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H1: Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nSet \\(G=S^*S\\), \\(\\tau=1/2\\), and define matrices \\(P,Q,Y\\) by\n\\[\nP_{ij}=\\frac{G_{ij}}{1-\\tau^2\\overline{\\lambda_i}\\lambda_j},\\qquad\nQ_{ij}=\\frac{G_{ij}}{1-\\tau\\overline{\\lambda_i}\\lambda_j},\\qquad Y=Q-P.\n\\]\nThen\n\\[\n4Y-YG^{-1}P-PG^{-1}Y\\succeq0.\n\\]\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, and \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\).",
    "starting_statement": null,
    "target_statement": "Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nSet \\(G=S^*S\\), \\(\\tau=1/2\\), and define matrices \\(P,Q,Y\\) by\n\\[\nP_{ij}=\\frac{G_{ij}}{1-\\tau^2\\overline{\\lambda_i}\\lambda_j},\\qquad\nQ_{ij}=\\frac{G_{ij}}{1-\\tau\\overline{\\lambda_i}\\lambda_j},\\qquad Y=Q-P.\n\\]\nThen\n\\[\n4Y-YG^{-1}P-PG^{-1}Y\\succeq0.\n\\]\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, and \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\).",
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  "OA-1-H0-to-H2": {
    "id": "OA-1-H0-to-H2",
    "problem_id": "OA-1",
    "problem_name": "Scalar Crouzeix inequality",
    "eval_version": "other-v5",
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    "start": 0,
    "target": 2,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H2: Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nThen \\(\\|T\\|\\le2\\).\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\), and \\(\\|T\\|\\) is its Euclidean operator norm.",
    "starting_statement": null,
    "target_statement": "Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nThen \\(\\|T\\|\\le2\\).\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\), and \\(\\|T\\|\\) is its Euclidean operator norm.",
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  "OA-1-H0-to-H3": {
    "id": "OA-1-H0-to-H3",
    "problem_id": "OA-1",
    "problem_name": "Scalar Crouzeix inequality",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H3: Let \\(\\Omega\\subset\\mathbb C\\) be a bounded convex domain with \\(C^2\\) boundary, and let \\(\\gamma:[0,L]\\to\\partial\\Omega\\) traverse the boundary once counterclockwise at unit speed. Put \\(\\nu(s)=-\\mathrm i\\gamma'(s)\\).\nLet \\(B\\in M_n(\\mathbb C)\\) satisfy\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\}\\subset\\Omega,\n\\]\nand define\n\\[\nR_s=(\\gamma(s)I-B)^{-1},\\qquad\nD_B(s)=\\frac1{2\\pi}\\left(\\nu(s)R_s+\\overline{\\nu(s)}R_s^*\\right).\n\\]\nThe map\n\\[\n\\Phi:C(\\partial\\Omega)\\to M_n(\\mathbb C),\\qquad\n\\Phi(h)=\\frac12\\int_0^L h(\\gamma(s))D_B(s)\\,ds\n\\]\nis positive and unital: \\(h\\ge0\\) implies \\(\\Phi(h)\\) is Hermitian positive semidefinite, and \\(\\Phi(1)=I\\).",
    "starting_statement": null,
    "target_statement": "Let \\(\\Omega\\subset\\mathbb C\\) be a bounded convex domain with \\(C^2\\) boundary, and let \\(\\gamma:[0,L]\\to\\partial\\Omega\\) traverse the boundary once counterclockwise at unit speed. Put \\(\\nu(s)=-\\mathrm i\\gamma'(s)\\).\nLet \\(B\\in M_n(\\mathbb C)\\) satisfy\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\}\\subset\\Omega,\n\\]\nand define\n\\[\nR_s=(\\gamma(s)I-B)^{-1},\\qquad\nD_B(s)=\\frac1{2\\pi}\\left(\\nu(s)R_s+\\overline{\\nu(s)}R_s^*\\right).\n\\]\nThe map\n\\[\n\\Phi:C(\\partial\\Omega)\\to M_n(\\mathbb C),\\qquad\n\\Phi(h)=\\frac12\\int_0^L h(\\gamma(s))D_B(s)\\,ds\n\\]\nis positive and unital: \\(h\\ge0\\) implies \\(\\Phi(h)\\) is Hermitian positive semidefinite, and \\(\\Phi(1)=I\\).",
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  "OA-1-H0-to-H4": {
    "id": "OA-1-H0-to-H4",
    "problem_id": "OA-1",
    "problem_name": "Scalar Crouzeix inequality",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H4: For every \\(n\\ge1\\), every \\(B\\in M_n(\\mathbb C)\\) with pairwise distinct eigenvalues, and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(B)\\|\\le2\\max_{z\\in W(B)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, and \\(f(B)\\) is defined by holomorphic functional calculus.",
    "starting_statement": null,
    "target_statement": "For every \\(n\\ge1\\), every \\(B\\in M_n(\\mathbb C)\\) with pairwise distinct eigenvalues, and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(B)\\|\\le2\\max_{z\\in W(B)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, and \\(f(B)\\) is defined by holomorphic functional calculus.",
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  "OA-1-H0-to-H5": {
    "id": "OA-1-H0-to-H5",
    "problem_id": "OA-1",
    "problem_name": "Scalar Crouzeix inequality",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H5: For every \\(n\\ge1\\), every \\(A\\in M_n(\\mathbb C)\\), and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(A)=\\{x^*Ax:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(A)\\|\\le2\\max_{z\\in W(A)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, \\(f(A)\\) is defined by holomorphic functional calculus, and the constant \\(2\\) is optimal uniformly over \\(n,A,f\\).",
    "starting_statement": null,
    "target_statement": "For every \\(n\\ge1\\), every \\(A\\in M_n(\\mathbb C)\\), and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(A)=\\{x^*Ax:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(A)\\|\\le2\\max_{z\\in W(A)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, \\(f(A)\\) is defined by holomorphic functional calculus, and the constant \\(2\\) is optimal uniformly over \\(n,A,f\\).",
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  "OA-1-H1-to-H2": {
    "id": "OA-1-H1-to-H2",
    "problem_id": "OA-1",
    "problem_name": "Scalar Crouzeix inequality",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 2,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nSet \\(G=S^*S\\), \\(\\tau=1/2\\), and define matrices \\(P,Q,Y\\) by\n\\[\nP_{ij}=\\frac{G_{ij}}{1-\\tau^2\\overline{\\lambda_i}\\lambda_j},\\qquad\nQ_{ij}=\\frac{G_{ij}}{1-\\tau\\overline{\\lambda_i}\\lambda_j},\\qquad Y=Q-P.\n\\]\nThen\n\\[\n4Y-YG^{-1}P-PG^{-1}Y\\succeq0.\n\\]\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, and \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\).\n\nProve H2: Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nThen \\(\\|T\\|\\le2\\).\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\), and \\(\\|T\\|\\) is its Euclidean operator norm.",
    "starting_statement": "Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nSet \\(G=S^*S\\), \\(\\tau=1/2\\), and define matrices \\(P,Q,Y\\) by\n\\[\nP_{ij}=\\frac{G_{ij}}{1-\\tau^2\\overline{\\lambda_i}\\lambda_j},\\qquad\nQ_{ij}=\\frac{G_{ij}}{1-\\tau\\overline{\\lambda_i}\\lambda_j},\\qquad Y=Q-P.\n\\]\nThen\n\\[\n4Y-YG^{-1}P-PG^{-1}Y\\succeq0.\n\\]\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, and \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\).",
    "target_statement": "Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nThen \\(\\|T\\|\\le2\\).\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\), and \\(\\|T\\|\\) is its Euclidean operator norm.",
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  "OA-1-H1-to-H3": {
    "id": "OA-1-H1-to-H3",
    "problem_id": "OA-1",
    "problem_name": "Scalar Crouzeix inequality",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nSet \\(G=S^*S\\), \\(\\tau=1/2\\), and define matrices \\(P,Q,Y\\) by\n\\[\nP_{ij}=\\frac{G_{ij}}{1-\\tau^2\\overline{\\lambda_i}\\lambda_j},\\qquad\nQ_{ij}=\\frac{G_{ij}}{1-\\tau\\overline{\\lambda_i}\\lambda_j},\\qquad Y=Q-P.\n\\]\nThen\n\\[\n4Y-YG^{-1}P-PG^{-1}Y\\succeq0.\n\\]\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, and \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\).\n\nProve H3: Let \\(\\Omega\\subset\\mathbb C\\) be a bounded convex domain with \\(C^2\\) boundary, and let \\(\\gamma:[0,L]\\to\\partial\\Omega\\) traverse the boundary once counterclockwise at unit speed. Put \\(\\nu(s)=-\\mathrm i\\gamma'(s)\\).\nLet \\(B\\in M_n(\\mathbb C)\\) satisfy\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\}\\subset\\Omega,\n\\]\nand define\n\\[\nR_s=(\\gamma(s)I-B)^{-1},\\qquad\nD_B(s)=\\frac1{2\\pi}\\left(\\nu(s)R_s+\\overline{\\nu(s)}R_s^*\\right).\n\\]\nThe map\n\\[\n\\Phi:C(\\partial\\Omega)\\to M_n(\\mathbb C),\\qquad\n\\Phi(h)=\\frac12\\int_0^L h(\\gamma(s))D_B(s)\\,ds\n\\]\nis positive and unital: \\(h\\ge0\\) implies \\(\\Phi(h)\\) is Hermitian positive semidefinite, and \\(\\Phi(1)=I\\).",
    "starting_statement": "Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nSet \\(G=S^*S\\), \\(\\tau=1/2\\), and define matrices \\(P,Q,Y\\) by\n\\[\nP_{ij}=\\frac{G_{ij}}{1-\\tau^2\\overline{\\lambda_i}\\lambda_j},\\qquad\nQ_{ij}=\\frac{G_{ij}}{1-\\tau\\overline{\\lambda_i}\\lambda_j},\\qquad Y=Q-P.\n\\]\nThen\n\\[\n4Y-YG^{-1}P-PG^{-1}Y\\succeq0.\n\\]\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, and \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\).",
    "target_statement": "Let \\(\\Omega\\subset\\mathbb C\\) be a bounded convex domain with \\(C^2\\) boundary, and let \\(\\gamma:[0,L]\\to\\partial\\Omega\\) traverse the boundary once counterclockwise at unit speed. Put \\(\\nu(s)=-\\mathrm i\\gamma'(s)\\).\nLet \\(B\\in M_n(\\mathbb C)\\) satisfy\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\}\\subset\\Omega,\n\\]\nand define\n\\[\nR_s=(\\gamma(s)I-B)^{-1},\\qquad\nD_B(s)=\\frac1{2\\pi}\\left(\\nu(s)R_s+\\overline{\\nu(s)}R_s^*\\right).\n\\]\nThe map\n\\[\n\\Phi:C(\\partial\\Omega)\\to M_n(\\mathbb C),\\qquad\n\\Phi(h)=\\frac12\\int_0^L h(\\gamma(s))D_B(s)\\,ds\n\\]\nis positive and unital: \\(h\\ge0\\) implies \\(\\Phi(h)\\) is Hermitian positive semidefinite, and \\(\\Phi(1)=I\\).",
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  "OA-1-H1-to-H4": {
    "id": "OA-1-H1-to-H4",
    "problem_id": "OA-1",
    "problem_name": "Scalar Crouzeix inequality",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nSet \\(G=S^*S\\), \\(\\tau=1/2\\), and define matrices \\(P,Q,Y\\) by\n\\[\nP_{ij}=\\frac{G_{ij}}{1-\\tau^2\\overline{\\lambda_i}\\lambda_j},\\qquad\nQ_{ij}=\\frac{G_{ij}}{1-\\tau\\overline{\\lambda_i}\\lambda_j},\\qquad Y=Q-P.\n\\]\nThen\n\\[\n4Y-YG^{-1}P-PG^{-1}Y\\succeq0.\n\\]\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, and \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\).\n\nProve H4: For every \\(n\\ge1\\), every \\(B\\in M_n(\\mathbb C)\\) with pairwise distinct eigenvalues, and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(B)\\|\\le2\\max_{z\\in W(B)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, and \\(f(B)\\) is defined by holomorphic functional calculus.",
    "starting_statement": "Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nSet \\(G=S^*S\\), \\(\\tau=1/2\\), and define matrices \\(P,Q,Y\\) by\n\\[\nP_{ij}=\\frac{G_{ij}}{1-\\tau^2\\overline{\\lambda_i}\\lambda_j},\\qquad\nQ_{ij}=\\frac{G_{ij}}{1-\\tau\\overline{\\lambda_i}\\lambda_j},\\qquad Y=Q-P.\n\\]\nThen\n\\[\n4Y-YG^{-1}P-PG^{-1}Y\\succeq0.\n\\]\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, and \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\).",
    "target_statement": "For every \\(n\\ge1\\), every \\(B\\in M_n(\\mathbb C)\\) with pairwise distinct eigenvalues, and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(B)\\|\\le2\\max_{z\\in W(B)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, and \\(f(B)\\) is defined by holomorphic functional calculus.",
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  "OA-1-H1-to-H5": {
    "id": "OA-1-H1-to-H5",
    "problem_id": "OA-1",
    "problem_name": "Scalar Crouzeix inequality",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nSet \\(G=S^*S\\), \\(\\tau=1/2\\), and define matrices \\(P,Q,Y\\) by\n\\[\nP_{ij}=\\frac{G_{ij}}{1-\\tau^2\\overline{\\lambda_i}\\lambda_j},\\qquad\nQ_{ij}=\\frac{G_{ij}}{1-\\tau\\overline{\\lambda_i}\\lambda_j},\\qquad Y=Q-P.\n\\]\nThen\n\\[\n4Y-YG^{-1}P-PG^{-1}Y\\succeq0.\n\\]\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, and \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\).\n\nProve H5: For every \\(n\\ge1\\), every \\(A\\in M_n(\\mathbb C)\\), and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(A)=\\{x^*Ax:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(A)\\|\\le2\\max_{z\\in W(A)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, \\(f(A)\\) is defined by holomorphic functional calculus, and the constant \\(2\\) is optimal uniformly over \\(n,A,f\\).",
    "starting_statement": "Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nSet \\(G=S^*S\\), \\(\\tau=1/2\\), and define matrices \\(P,Q,Y\\) by\n\\[\nP_{ij}=\\frac{G_{ij}}{1-\\tau^2\\overline{\\lambda_i}\\lambda_j},\\qquad\nQ_{ij}=\\frac{G_{ij}}{1-\\tau\\overline{\\lambda_i}\\lambda_j},\\qquad Y=Q-P.\n\\]\nThen\n\\[\n4Y-YG^{-1}P-PG^{-1}Y\\succeq0.\n\\]\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, and \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\).",
    "target_statement": "For every \\(n\\ge1\\), every \\(A\\in M_n(\\mathbb C)\\), and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(A)=\\{x^*Ax:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(A)\\|\\le2\\max_{z\\in W(A)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, \\(f(A)\\) is defined by holomorphic functional calculus, and the constant \\(2\\) is optimal uniformly over \\(n,A,f\\).",
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  "OA-1-H2-to-H3": {
    "id": "OA-1-H2-to-H3",
    "problem_id": "OA-1",
    "problem_name": "Scalar Crouzeix inequality",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nThen \\(\\|T\\|\\le2\\).\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\), and \\(\\|T\\|\\) is its Euclidean operator norm.\n\nProve H3: Let \\(\\Omega\\subset\\mathbb C\\) be a bounded convex domain with \\(C^2\\) boundary, and let \\(\\gamma:[0,L]\\to\\partial\\Omega\\) traverse the boundary once counterclockwise at unit speed. Put \\(\\nu(s)=-\\mathrm i\\gamma'(s)\\).\nLet \\(B\\in M_n(\\mathbb C)\\) satisfy\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\}\\subset\\Omega,\n\\]\nand define\n\\[\nR_s=(\\gamma(s)I-B)^{-1},\\qquad\nD_B(s)=\\frac1{2\\pi}\\left(\\nu(s)R_s+\\overline{\\nu(s)}R_s^*\\right).\n\\]\nThe map\n\\[\n\\Phi:C(\\partial\\Omega)\\to M_n(\\mathbb C),\\qquad\n\\Phi(h)=\\frac12\\int_0^L h(\\gamma(s))D_B(s)\\,ds\n\\]\nis positive and unital: \\(h\\ge0\\) implies \\(\\Phi(h)\\) is Hermitian positive semidefinite, and \\(\\Phi(1)=I\\).",
    "starting_statement": "Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nThen \\(\\|T\\|\\le2\\).\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\), and \\(\\|T\\|\\) is its Euclidean operator norm.",
    "target_statement": "Let \\(\\Omega\\subset\\mathbb C\\) be a bounded convex domain with \\(C^2\\) boundary, and let \\(\\gamma:[0,L]\\to\\partial\\Omega\\) traverse the boundary once counterclockwise at unit speed. Put \\(\\nu(s)=-\\mathrm i\\gamma'(s)\\).\nLet \\(B\\in M_n(\\mathbb C)\\) satisfy\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\}\\subset\\Omega,\n\\]\nand define\n\\[\nR_s=(\\gamma(s)I-B)^{-1},\\qquad\nD_B(s)=\\frac1{2\\pi}\\left(\\nu(s)R_s+\\overline{\\nu(s)}R_s^*\\right).\n\\]\nThe map\n\\[\n\\Phi:C(\\partial\\Omega)\\to M_n(\\mathbb C),\\qquad\n\\Phi(h)=\\frac12\\int_0^L h(\\gamma(s))D_B(s)\\,ds\n\\]\nis positive and unital: \\(h\\ge0\\) implies \\(\\Phi(h)\\) is Hermitian positive semidefinite, and \\(\\Phi(1)=I\\).",
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  "OA-1-H2-to-H4": {
    "id": "OA-1-H2-to-H4",
    "problem_id": "OA-1",
    "problem_name": "Scalar Crouzeix inequality",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nThen \\(\\|T\\|\\le2\\).\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\), and \\(\\|T\\|\\) is its Euclidean operator norm.\n\nProve H4: For every \\(n\\ge1\\), every \\(B\\in M_n(\\mathbb C)\\) with pairwise distinct eigenvalues, and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(B)\\|\\le2\\max_{z\\in W(B)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, and \\(f(B)\\) is defined by holomorphic functional calculus.",
    "starting_statement": "Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nThen \\(\\|T\\|\\le2\\).\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\), and \\(\\|T\\|\\) is its Euclidean operator norm.",
    "target_statement": "For every \\(n\\ge1\\), every \\(B\\in M_n(\\mathbb C)\\) with pairwise distinct eigenvalues, and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(B)\\|\\le2\\max_{z\\in W(B)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, and \\(f(B)\\) is defined by holomorphic functional calculus.",
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  "OA-1-H2-to-H5": {
    "id": "OA-1-H2-to-H5",
    "problem_id": "OA-1",
    "problem_name": "Scalar Crouzeix inequality",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nThen \\(\\|T\\|\\le2\\).\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\), and \\(\\|T\\|\\) is its Euclidean operator norm.\n\nProve H5: For every \\(n\\ge1\\), every \\(A\\in M_n(\\mathbb C)\\), and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(A)=\\{x^*Ax:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(A)\\|\\le2\\max_{z\\in W(A)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, \\(f(A)\\) is defined by holomorphic functional calculus, and the constant \\(2\\) is optimal uniformly over \\(n,A,f\\).",
    "starting_statement": "Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nThen \\(\\|T\\|\\le2\\).\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\), and \\(\\|T\\|\\) is its Euclidean operator norm.",
    "target_statement": "For every \\(n\\ge1\\), every \\(A\\in M_n(\\mathbb C)\\), and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(A)=\\{x^*Ax:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(A)\\|\\le2\\max_{z\\in W(A)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, \\(f(A)\\) is defined by holomorphic functional calculus, and the constant \\(2\\) is optimal uniformly over \\(n,A,f\\).",
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  "OA-1-H3-to-H4": {
    "id": "OA-1-H3-to-H4",
    "problem_id": "OA-1",
    "problem_name": "Scalar Crouzeix inequality",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H3: Let \\(\\Omega\\subset\\mathbb C\\) be a bounded convex domain with \\(C^2\\) boundary, and let \\(\\gamma:[0,L]\\to\\partial\\Omega\\) traverse the boundary once counterclockwise at unit speed. Put \\(\\nu(s)=-\\mathrm i\\gamma'(s)\\).\nLet \\(B\\in M_n(\\mathbb C)\\) satisfy\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\}\\subset\\Omega,\n\\]\nand define\n\\[\nR_s=(\\gamma(s)I-B)^{-1},\\qquad\nD_B(s)=\\frac1{2\\pi}\\left(\\nu(s)R_s+\\overline{\\nu(s)}R_s^*\\right).\n\\]\nThe map\n\\[\n\\Phi:C(\\partial\\Omega)\\to M_n(\\mathbb C),\\qquad\n\\Phi(h)=\\frac12\\int_0^L h(\\gamma(s))D_B(s)\\,ds\n\\]\nis positive and unital: \\(h\\ge0\\) implies \\(\\Phi(h)\\) is Hermitian positive semidefinite, and \\(\\Phi(1)=I\\).\n\nProve H4: For every \\(n\\ge1\\), every \\(B\\in M_n(\\mathbb C)\\) with pairwise distinct eigenvalues, and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(B)\\|\\le2\\max_{z\\in W(B)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, and \\(f(B)\\) is defined by holomorphic functional calculus.",
    "starting_statement": "Let \\(\\Omega\\subset\\mathbb C\\) be a bounded convex domain with \\(C^2\\) boundary, and let \\(\\gamma:[0,L]\\to\\partial\\Omega\\) traverse the boundary once counterclockwise at unit speed. Put \\(\\nu(s)=-\\mathrm i\\gamma'(s)\\).\nLet \\(B\\in M_n(\\mathbb C)\\) satisfy\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\}\\subset\\Omega,\n\\]\nand define\n\\[\nR_s=(\\gamma(s)I-B)^{-1},\\qquad\nD_B(s)=\\frac1{2\\pi}\\left(\\nu(s)R_s+\\overline{\\nu(s)}R_s^*\\right).\n\\]\nThe map\n\\[\n\\Phi:C(\\partial\\Omega)\\to M_n(\\mathbb C),\\qquad\n\\Phi(h)=\\frac12\\int_0^L h(\\gamma(s))D_B(s)\\,ds\n\\]\nis positive and unital: \\(h\\ge0\\) implies \\(\\Phi(h)\\) is Hermitian positive semidefinite, and \\(\\Phi(1)=I\\).",
    "target_statement": "For every \\(n\\ge1\\), every \\(B\\in M_n(\\mathbb C)\\) with pairwise distinct eigenvalues, and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(B)\\|\\le2\\max_{z\\in W(B)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, and \\(f(B)\\) is defined by holomorphic functional calculus.",
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  "OA-1-H3-to-H5": {
    "id": "OA-1-H3-to-H5",
    "problem_id": "OA-1",
    "problem_name": "Scalar Crouzeix inequality",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H3: Let \\(\\Omega\\subset\\mathbb C\\) be a bounded convex domain with \\(C^2\\) boundary, and let \\(\\gamma:[0,L]\\to\\partial\\Omega\\) traverse the boundary once counterclockwise at unit speed. Put \\(\\nu(s)=-\\mathrm i\\gamma'(s)\\).\nLet \\(B\\in M_n(\\mathbb C)\\) satisfy\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\}\\subset\\Omega,\n\\]\nand define\n\\[\nR_s=(\\gamma(s)I-B)^{-1},\\qquad\nD_B(s)=\\frac1{2\\pi}\\left(\\nu(s)R_s+\\overline{\\nu(s)}R_s^*\\right).\n\\]\nThe map\n\\[\n\\Phi:C(\\partial\\Omega)\\to M_n(\\mathbb C),\\qquad\n\\Phi(h)=\\frac12\\int_0^L h(\\gamma(s))D_B(s)\\,ds\n\\]\nis positive and unital: \\(h\\ge0\\) implies \\(\\Phi(h)\\) is Hermitian positive semidefinite, and \\(\\Phi(1)=I\\).\n\nProve H5: For every \\(n\\ge1\\), every \\(A\\in M_n(\\mathbb C)\\), and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(A)=\\{x^*Ax:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(A)\\|\\le2\\max_{z\\in W(A)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, \\(f(A)\\) is defined by holomorphic functional calculus, and the constant \\(2\\) is optimal uniformly over \\(n,A,f\\).",
    "starting_statement": "Let \\(\\Omega\\subset\\mathbb C\\) be a bounded convex domain with \\(C^2\\) boundary, and let \\(\\gamma:[0,L]\\to\\partial\\Omega\\) traverse the boundary once counterclockwise at unit speed. Put \\(\\nu(s)=-\\mathrm i\\gamma'(s)\\).\nLet \\(B\\in M_n(\\mathbb C)\\) satisfy\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\}\\subset\\Omega,\n\\]\nand define\n\\[\nR_s=(\\gamma(s)I-B)^{-1},\\qquad\nD_B(s)=\\frac1{2\\pi}\\left(\\nu(s)R_s+\\overline{\\nu(s)}R_s^*\\right).\n\\]\nThe map\n\\[\n\\Phi:C(\\partial\\Omega)\\to M_n(\\mathbb C),\\qquad\n\\Phi(h)=\\frac12\\int_0^L h(\\gamma(s))D_B(s)\\,ds\n\\]\nis positive and unital: \\(h\\ge0\\) implies \\(\\Phi(h)\\) is Hermitian positive semidefinite, and \\(\\Phi(1)=I\\).",
    "target_statement": "For every \\(n\\ge1\\), every \\(A\\in M_n(\\mathbb C)\\), and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(A)=\\{x^*Ax:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(A)\\|\\le2\\max_{z\\in W(A)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, \\(f(A)\\) is defined by holomorphic functional calculus, and the constant \\(2\\) is optimal uniformly over \\(n,A,f\\).",
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  },
  "OA-1-H4-to-H5": {
    "id": "OA-1-H4-to-H5",
    "problem_id": "OA-1",
    "problem_name": "Scalar Crouzeix inequality",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 4,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H4: For every \\(n\\ge1\\), every \\(B\\in M_n(\\mathbb C)\\) with pairwise distinct eigenvalues, and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(B)\\|\\le2\\max_{z\\in W(B)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, and \\(f(B)\\) is defined by holomorphic functional calculus.\n\nProve H5: For every \\(n\\ge1\\), every \\(A\\in M_n(\\mathbb C)\\), and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(A)=\\{x^*Ax:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(A)\\|\\le2\\max_{z\\in W(A)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, \\(f(A)\\) is defined by holomorphic functional calculus, and the constant \\(2\\) is optimal uniformly over \\(n,A,f\\).",
    "starting_statement": "For every \\(n\\ge1\\), every \\(B\\in M_n(\\mathbb C)\\) with pairwise distinct eigenvalues, and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(B)\\|\\le2\\max_{z\\in W(B)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, and \\(f(B)\\) is defined by holomorphic functional calculus.",
    "target_statement": "For every \\(n\\ge1\\), every \\(A\\in M_n(\\mathbb C)\\), and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(A)=\\{x^*Ax:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(A)\\|\\le2\\max_{z\\in W(A)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, \\(f(A)\\) is defined by holomorphic functional calculus, and the constant \\(2\\) is optimal uniformly over \\(n,A,f\\).",
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  "OA-2-H0-to-H1": {
    "id": "OA-2-H0-to-H1",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 1,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H1: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nEach \\(\\theta_i\\) is an action of \\(H\\) by additive automorphisms of \\(D\\), and the canonical maps from \\(D\\) and \\(H\\) give an isomorphism\n\\[\nD\\rtimes_{\\theta_i}H\\cong\\Gamma_i\\qquad(i=1,2).\n\\]",
    "starting_statement": null,
    "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nEach \\(\\theta_i\\) is an action of \\(H\\) by additive automorphisms of \\(D\\), and the canonical maps from \\(D\\) and \\(H\\) give an isomorphism\n\\[\nD\\rtimes_{\\theta_i}H\\cong\\Gamma_i\\qquad(i=1,2).\n\\]",
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  "OA-2-H0-to-H2": {
    "id": "OA-2-H0-to-H2",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 2,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H2: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThere is a trace-preserving normal \\(*\\)-isomorphism\n\\[\nL(\\Gamma_1)\\cong L(\\Gamma_2).\n\\]\nFor a discrete group \\(\\Gamma\\), \\(L(\\Gamma)\\) is generated by its left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\).",
    "starting_statement": null,
    "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThere is a trace-preserving normal \\(*\\)-isomorphism\n\\[\nL(\\Gamma_1)\\cong L(\\Gamma_2).\n\\]\nFor a discrete group \\(\\Gamma\\), \\(L(\\Gamma)\\) is generated by its left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\).",
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  "OA-2-H0-to-H3": {
    "id": "OA-2-H0-to-H3",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H3: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]",
    "starting_statement": null,
    "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]",
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  "OA-2-H0-to-H4": {
    "id": "OA-2-H0-to-H4",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H4: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nBoth \\(\\Gamma_1\\) and \\(\\Gamma_2\\) have property \\((T)\\): every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that, for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), some unit vector \\(\\xi\\) satisfies \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).",
    "starting_statement": null,
    "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nBoth \\(\\Gamma_1\\) and \\(\\Gamma_2\\) have property \\((T)\\): every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that, for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), some unit vector \\(\\xi\\) satisfies \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).",
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  },
  "OA-2-H0-to-H5": {
    "id": "OA-2-H0-to-H5",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H5: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThe groups are not isomorphic as abstract groups:\n\\[\n\\Gamma_1\\not\\cong\\Gamma_2.\n\\]",
    "starting_statement": null,
    "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThe groups are not isomorphic as abstract groups:\n\\[\n\\Gamma_1\\not\\cong\\Gamma_2.\n\\]",
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  },
  "OA-2-H0-to-H6": {
    "id": "OA-2-H0-to-H6",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H6: A discrete group has property \\((T)\\) if every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), there is a unit vector \\(\\xi\\) with \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\nA group is ICC if the conjugacy class of every nonidentity element is infinite.\n\nThere exist two countable ICC groups \\(\\Gamma\\) and \\(\\Lambda\\), both with property \\((T)\\), which are not isomorphic as abstract groups, but whose group von Neumann algebras are trace-preservingly normally \\(*\\)-isomorphic \\({\\rm II}_1\\) factors:\n\\[\n\\Gamma\\not\\cong\\Lambda,\\qquad L(\\Gamma)\\cong L(\\Lambda).\n\\]\nThe algebra \\(L(\\Gamma)\\) is generated by the left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\); \\(L(\\Lambda)\\) is defined in the same way. A \\({\\rm II}_1\\) factor is an infinite-dimensional von Neumann algebra with center \\(\\mathbb C I\\) and a faithful normal tracial state.",
    "starting_statement": null,
    "target_statement": "A discrete group has property \\((T)\\) if every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), there is a unit vector \\(\\xi\\) with \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\nA group is ICC if the conjugacy class of every nonidentity element is infinite.\n\nThere exist two countable ICC groups \\(\\Gamma\\) and \\(\\Lambda\\), both with property \\((T)\\), which are not isomorphic as abstract groups, but whose group von Neumann algebras are trace-preservingly normally \\(*\\)-isomorphic \\({\\rm II}_1\\) factors:\n\\[\n\\Gamma\\not\\cong\\Lambda,\\qquad L(\\Gamma)\\cong L(\\Lambda).\n\\]\nThe algebra \\(L(\\Gamma)\\) is generated by the left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\); \\(L(\\Lambda)\\) is defined in the same way. A \\({\\rm II}_1\\) factor is an infinite-dimensional von Neumann algebra with center \\(\\mathbb C I\\) and a faithful normal tracial state.",
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  "OA-2-H1-to-H2": {
    "id": "OA-2-H1-to-H2",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 2,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nEach \\(\\theta_i\\) is an action of \\(H\\) by additive automorphisms of \\(D\\), and the canonical maps from \\(D\\) and \\(H\\) give an isomorphism\n\\[\nD\\rtimes_{\\theta_i}H\\cong\\Gamma_i\\qquad(i=1,2).\n\\]\n\nProve H2: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThere is a trace-preserving normal \\(*\\)-isomorphism\n\\[\nL(\\Gamma_1)\\cong L(\\Gamma_2).\n\\]\nFor a discrete group \\(\\Gamma\\), \\(L(\\Gamma)\\) is generated by its left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\).",
    "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nEach \\(\\theta_i\\) is an action of \\(H\\) by additive automorphisms of \\(D\\), and the canonical maps from \\(D\\) and \\(H\\) give an isomorphism\n\\[\nD\\rtimes_{\\theta_i}H\\cong\\Gamma_i\\qquad(i=1,2).\n\\]",
    "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThere is a trace-preserving normal \\(*\\)-isomorphism\n\\[\nL(\\Gamma_1)\\cong L(\\Gamma_2).\n\\]\nFor a discrete group \\(\\Gamma\\), \\(L(\\Gamma)\\) is generated by its left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\).",
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  "OA-2-H1-to-H3": {
    "id": "OA-2-H1-to-H3",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nEach \\(\\theta_i\\) is an action of \\(H\\) by additive automorphisms of \\(D\\), and the canonical maps from \\(D\\) and \\(H\\) give an isomorphism\n\\[\nD\\rtimes_{\\theta_i}H\\cong\\Gamma_i\\qquad(i=1,2).\n\\]\n\nProve H3: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]",
    "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nEach \\(\\theta_i\\) is an action of \\(H\\) by additive automorphisms of \\(D\\), and the canonical maps from \\(D\\) and \\(H\\) give an isomorphism\n\\[\nD\\rtimes_{\\theta_i}H\\cong\\Gamma_i\\qquad(i=1,2).\n\\]",
    "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]",
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  "OA-2-H1-to-H4": {
    "id": "OA-2-H1-to-H4",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nEach \\(\\theta_i\\) is an action of \\(H\\) by additive automorphisms of \\(D\\), and the canonical maps from \\(D\\) and \\(H\\) give an isomorphism\n\\[\nD\\rtimes_{\\theta_i}H\\cong\\Gamma_i\\qquad(i=1,2).\n\\]\n\nProve H4: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nBoth \\(\\Gamma_1\\) and \\(\\Gamma_2\\) have property \\((T)\\): every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that, for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), some unit vector \\(\\xi\\) satisfies \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).",
    "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nEach \\(\\theta_i\\) is an action of \\(H\\) by additive automorphisms of \\(D\\), and the canonical maps from \\(D\\) and \\(H\\) give an isomorphism\n\\[\nD\\rtimes_{\\theta_i}H\\cong\\Gamma_i\\qquad(i=1,2).\n\\]",
    "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nBoth \\(\\Gamma_1\\) and \\(\\Gamma_2\\) have property \\((T)\\): every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that, for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), some unit vector \\(\\xi\\) satisfies \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).",
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  "OA-2-H1-to-H5": {
    "id": "OA-2-H1-to-H5",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nEach \\(\\theta_i\\) is an action of \\(H\\) by additive automorphisms of \\(D\\), and the canonical maps from \\(D\\) and \\(H\\) give an isomorphism\n\\[\nD\\rtimes_{\\theta_i}H\\cong\\Gamma_i\\qquad(i=1,2).\n\\]\n\nProve H5: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThe groups are not isomorphic as abstract groups:\n\\[\n\\Gamma_1\\not\\cong\\Gamma_2.\n\\]",
    "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nEach \\(\\theta_i\\) is an action of \\(H\\) by additive automorphisms of \\(D\\), and the canonical maps from \\(D\\) and \\(H\\) give an isomorphism\n\\[\nD\\rtimes_{\\theta_i}H\\cong\\Gamma_i\\qquad(i=1,2).\n\\]",
    "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThe groups are not isomorphic as abstract groups:\n\\[\n\\Gamma_1\\not\\cong\\Gamma_2.\n\\]",
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  "OA-2-H1-to-H6": {
    "id": "OA-2-H1-to-H6",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nEach \\(\\theta_i\\) is an action of \\(H\\) by additive automorphisms of \\(D\\), and the canonical maps from \\(D\\) and \\(H\\) give an isomorphism\n\\[\nD\\rtimes_{\\theta_i}H\\cong\\Gamma_i\\qquad(i=1,2).\n\\]\n\nProve H6: A discrete group has property \\((T)\\) if every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), there is a unit vector \\(\\xi\\) with \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\nA group is ICC if the conjugacy class of every nonidentity element is infinite.\n\nThere exist two countable ICC groups \\(\\Gamma\\) and \\(\\Lambda\\), both with property \\((T)\\), which are not isomorphic as abstract groups, but whose group von Neumann algebras are trace-preservingly normally \\(*\\)-isomorphic \\({\\rm II}_1\\) factors:\n\\[\n\\Gamma\\not\\cong\\Lambda,\\qquad L(\\Gamma)\\cong L(\\Lambda).\n\\]\nThe algebra \\(L(\\Gamma)\\) is generated by the left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\); \\(L(\\Lambda)\\) is defined in the same way. A \\({\\rm II}_1\\) factor is an infinite-dimensional von Neumann algebra with center \\(\\mathbb C I\\) and a faithful normal tracial state.",
    "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nEach \\(\\theta_i\\) is an action of \\(H\\) by additive automorphisms of \\(D\\), and the canonical maps from \\(D\\) and \\(H\\) give an isomorphism\n\\[\nD\\rtimes_{\\theta_i}H\\cong\\Gamma_i\\qquad(i=1,2).\n\\]",
    "target_statement": "A discrete group has property \\((T)\\) if every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), there is a unit vector \\(\\xi\\) with \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\nA group is ICC if the conjugacy class of every nonidentity element is infinite.\n\nThere exist two countable ICC groups \\(\\Gamma\\) and \\(\\Lambda\\), both with property \\((T)\\), which are not isomorphic as abstract groups, but whose group von Neumann algebras are trace-preservingly normally \\(*\\)-isomorphic \\({\\rm II}_1\\) factors:\n\\[\n\\Gamma\\not\\cong\\Lambda,\\qquad L(\\Gamma)\\cong L(\\Lambda).\n\\]\nThe algebra \\(L(\\Gamma)\\) is generated by the left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\); \\(L(\\Lambda)\\) is defined in the same way. A \\({\\rm II}_1\\) factor is an infinite-dimensional von Neumann algebra with center \\(\\mathbb C I\\) and a faithful normal tracial state.",
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  "OA-2-H2-to-H3": {
    "id": "OA-2-H2-to-H3",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThere is a trace-preserving normal \\(*\\)-isomorphism\n\\[\nL(\\Gamma_1)\\cong L(\\Gamma_2).\n\\]\nFor a discrete group \\(\\Gamma\\), \\(L(\\Gamma)\\) is generated by its left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\).\n\nProve H3: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]",
    "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThere is a trace-preserving normal \\(*\\)-isomorphism\n\\[\nL(\\Gamma_1)\\cong L(\\Gamma_2).\n\\]\nFor a discrete group \\(\\Gamma\\), \\(L(\\Gamma)\\) is generated by its left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\).",
    "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]",
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  "OA-2-H2-to-H4": {
    "id": "OA-2-H2-to-H4",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThere is a trace-preserving normal \\(*\\)-isomorphism\n\\[\nL(\\Gamma_1)\\cong L(\\Gamma_2).\n\\]\nFor a discrete group \\(\\Gamma\\), \\(L(\\Gamma)\\) is generated by its left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\).\n\nProve H4: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nBoth \\(\\Gamma_1\\) and \\(\\Gamma_2\\) have property \\((T)\\): every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that, for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), some unit vector \\(\\xi\\) satisfies \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).",
    "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThere is a trace-preserving normal \\(*\\)-isomorphism\n\\[\nL(\\Gamma_1)\\cong L(\\Gamma_2).\n\\]\nFor a discrete group \\(\\Gamma\\), \\(L(\\Gamma)\\) is generated by its left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\).",
    "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nBoth \\(\\Gamma_1\\) and \\(\\Gamma_2\\) have property \\((T)\\): every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that, for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), some unit vector \\(\\xi\\) satisfies \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).",
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  "OA-2-H2-to-H5": {
    "id": "OA-2-H2-to-H5",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThere is a trace-preserving normal \\(*\\)-isomorphism\n\\[\nL(\\Gamma_1)\\cong L(\\Gamma_2).\n\\]\nFor a discrete group \\(\\Gamma\\), \\(L(\\Gamma)\\) is generated by its left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\).\n\nProve H5: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThe groups are not isomorphic as abstract groups:\n\\[\n\\Gamma_1\\not\\cong\\Gamma_2.\n\\]",
    "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThere is a trace-preserving normal \\(*\\)-isomorphism\n\\[\nL(\\Gamma_1)\\cong L(\\Gamma_2).\n\\]\nFor a discrete group \\(\\Gamma\\), \\(L(\\Gamma)\\) is generated by its left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\).",
    "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThe groups are not isomorphic as abstract groups:\n\\[\n\\Gamma_1\\not\\cong\\Gamma_2.\n\\]",
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  "OA-2-H2-to-H6": {
    "id": "OA-2-H2-to-H6",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThere is a trace-preserving normal \\(*\\)-isomorphism\n\\[\nL(\\Gamma_1)\\cong L(\\Gamma_2).\n\\]\nFor a discrete group \\(\\Gamma\\), \\(L(\\Gamma)\\) is generated by its left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\).\n\nProve H6: A discrete group has property \\((T)\\) if every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), there is a unit vector \\(\\xi\\) with \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\nA group is ICC if the conjugacy class of every nonidentity element is infinite.\n\nThere exist two countable ICC groups \\(\\Gamma\\) and \\(\\Lambda\\), both with property \\((T)\\), which are not isomorphic as abstract groups, but whose group von Neumann algebras are trace-preservingly normally \\(*\\)-isomorphic \\({\\rm II}_1\\) factors:\n\\[\n\\Gamma\\not\\cong\\Lambda,\\qquad L(\\Gamma)\\cong L(\\Lambda).\n\\]\nThe algebra \\(L(\\Gamma)\\) is generated by the left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\); \\(L(\\Lambda)\\) is defined in the same way. A \\({\\rm II}_1\\) factor is an infinite-dimensional von Neumann algebra with center \\(\\mathbb C I\\) and a faithful normal tracial state.",
    "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThere is a trace-preserving normal \\(*\\)-isomorphism\n\\[\nL(\\Gamma_1)\\cong L(\\Gamma_2).\n\\]\nFor a discrete group \\(\\Gamma\\), \\(L(\\Gamma)\\) is generated by its left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\).",
    "target_statement": "A discrete group has property \\((T)\\) if every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), there is a unit vector \\(\\xi\\) with \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\nA group is ICC if the conjugacy class of every nonidentity element is infinite.\n\nThere exist two countable ICC groups \\(\\Gamma\\) and \\(\\Lambda\\), both with property \\((T)\\), which are not isomorphic as abstract groups, but whose group von Neumann algebras are trace-preservingly normally \\(*\\)-isomorphic \\({\\rm II}_1\\) factors:\n\\[\n\\Gamma\\not\\cong\\Lambda,\\qquad L(\\Gamma)\\cong L(\\Lambda).\n\\]\nThe algebra \\(L(\\Gamma)\\) is generated by the left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\); \\(L(\\Lambda)\\) is defined in the same way. A \\({\\rm II}_1\\) factor is an infinite-dimensional von Neumann algebra with center \\(\\mathbb C I\\) and a faithful normal tracial state.",
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  "OA-2-H3-to-H4": {
    "id": "OA-2-H3-to-H4",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H3: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]\n\nProve H4: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nBoth \\(\\Gamma_1\\) and \\(\\Gamma_2\\) have property \\((T)\\): every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that, for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), some unit vector \\(\\xi\\) satisfies \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).",
    "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]",
    "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nBoth \\(\\Gamma_1\\) and \\(\\Gamma_2\\) have property \\((T)\\): every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that, for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), some unit vector \\(\\xi\\) satisfies \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).",
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  "OA-2-H3-to-H5": {
    "id": "OA-2-H3-to-H5",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H3: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]\n\nProve H5: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThe groups are not isomorphic as abstract groups:\n\\[\n\\Gamma_1\\not\\cong\\Gamma_2.\n\\]",
    "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]",
    "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThe groups are not isomorphic as abstract groups:\n\\[\n\\Gamma_1\\not\\cong\\Gamma_2.\n\\]",
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  "OA-2-H3-to-H6": {
    "id": "OA-2-H3-to-H6",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H3: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]\n\nProve H6: A discrete group has property \\((T)\\) if every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), there is a unit vector \\(\\xi\\) with \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\nA group is ICC if the conjugacy class of every nonidentity element is infinite.\n\nThere exist two countable ICC groups \\(\\Gamma\\) and \\(\\Lambda\\), both with property \\((T)\\), which are not isomorphic as abstract groups, but whose group von Neumann algebras are trace-preservingly normally \\(*\\)-isomorphic \\({\\rm II}_1\\) factors:\n\\[\n\\Gamma\\not\\cong\\Lambda,\\qquad L(\\Gamma)\\cong L(\\Lambda).\n\\]\nThe algebra \\(L(\\Gamma)\\) is generated by the left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\); \\(L(\\Lambda)\\) is defined in the same way. A \\({\\rm II}_1\\) factor is an infinite-dimensional von Neumann algebra with center \\(\\mathbb C I\\) and a faithful normal tracial state.",
    "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]",
    "target_statement": "A discrete group has property \\((T)\\) if every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), there is a unit vector \\(\\xi\\) with \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\nA group is ICC if the conjugacy class of every nonidentity element is infinite.\n\nThere exist two countable ICC groups \\(\\Gamma\\) and \\(\\Lambda\\), both with property \\((T)\\), which are not isomorphic as abstract groups, but whose group von Neumann algebras are trace-preservingly normally \\(*\\)-isomorphic \\({\\rm II}_1\\) factors:\n\\[\n\\Gamma\\not\\cong\\Lambda,\\qquad L(\\Gamma)\\cong L(\\Lambda).\n\\]\nThe algebra \\(L(\\Gamma)\\) is generated by the left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\); \\(L(\\Lambda)\\) is defined in the same way. A \\({\\rm II}_1\\) factor is an infinite-dimensional von Neumann algebra with center \\(\\mathbb C I\\) and a faithful normal tracial state.",
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  "OA-2-H4-to-H5": {
    "id": "OA-2-H4-to-H5",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 4,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H4: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nBoth \\(\\Gamma_1\\) and \\(\\Gamma_2\\) have property \\((T)\\): every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that, for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), some unit vector \\(\\xi\\) satisfies \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\n\nProve H5: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThe groups are not isomorphic as abstract groups:\n\\[\n\\Gamma_1\\not\\cong\\Gamma_2.\n\\]",
    "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nBoth \\(\\Gamma_1\\) and \\(\\Gamma_2\\) have property \\((T)\\): every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that, for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), some unit vector \\(\\xi\\) satisfies \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).",
    "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThe groups are not isomorphic as abstract groups:\n\\[\n\\Gamma_1\\not\\cong\\Gamma_2.\n\\]",
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  "OA-2-H4-to-H6": {
    "id": "OA-2-H4-to-H6",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 4,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H4: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nBoth \\(\\Gamma_1\\) and \\(\\Gamma_2\\) have property \\((T)\\): every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that, for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), some unit vector \\(\\xi\\) satisfies \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\n\nProve H6: A discrete group has property \\((T)\\) if every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), there is a unit vector \\(\\xi\\) with \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\nA group is ICC if the conjugacy class of every nonidentity element is infinite.\n\nThere exist two countable ICC groups \\(\\Gamma\\) and \\(\\Lambda\\), both with property \\((T)\\), which are not isomorphic as abstract groups, but whose group von Neumann algebras are trace-preservingly normally \\(*\\)-isomorphic \\({\\rm II}_1\\) factors:\n\\[\n\\Gamma\\not\\cong\\Lambda,\\qquad L(\\Gamma)\\cong L(\\Lambda).\n\\]\nThe algebra \\(L(\\Gamma)\\) is generated by the left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\); \\(L(\\Lambda)\\) is defined in the same way. A \\({\\rm II}_1\\) factor is an infinite-dimensional von Neumann algebra with center \\(\\mathbb C I\\) and a faithful normal tracial state.",
    "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nBoth \\(\\Gamma_1\\) and \\(\\Gamma_2\\) have property \\((T)\\): every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that, for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), some unit vector \\(\\xi\\) satisfies \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).",
    "target_statement": "A discrete group has property \\((T)\\) if every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), there is a unit vector \\(\\xi\\) with \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\nA group is ICC if the conjugacy class of every nonidentity element is infinite.\n\nThere exist two countable ICC groups \\(\\Gamma\\) and \\(\\Lambda\\), both with property \\((T)\\), which are not isomorphic as abstract groups, but whose group von Neumann algebras are trace-preservingly normally \\(*\\)-isomorphic \\({\\rm II}_1\\) factors:\n\\[\n\\Gamma\\not\\cong\\Lambda,\\qquad L(\\Gamma)\\cong L(\\Lambda).\n\\]\nThe algebra \\(L(\\Gamma)\\) is generated by the left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\); \\(L(\\Lambda)\\) is defined in the same way. A \\({\\rm II}_1\\) factor is an infinite-dimensional von Neumann algebra with center \\(\\mathbb C I\\) and a faithful normal tracial state.",
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  "OA-2-H5-to-H6": {
    "id": "OA-2-H5-to-H6",
    "problem_id": "OA-2",
    "problem_name": "Groups with isomorphic factors",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 5,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H5: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThe groups are not isomorphic as abstract groups:\n\\[\n\\Gamma_1\\not\\cong\\Gamma_2.\n\\]\n\nProve H6: A discrete group has property \\((T)\\) if every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), there is a unit vector \\(\\xi\\) with \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\nA group is ICC if the conjugacy class of every nonidentity element is infinite.\n\nThere exist two countable ICC groups \\(\\Gamma\\) and \\(\\Lambda\\), both with property \\((T)\\), which are not isomorphic as abstract groups, but whose group von Neumann algebras are trace-preservingly normally \\(*\\)-isomorphic \\({\\rm II}_1\\) factors:\n\\[\n\\Gamma\\not\\cong\\Lambda,\\qquad L(\\Gamma)\\cong L(\\Lambda).\n\\]\nThe algebra \\(L(\\Gamma)\\) is generated by the left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\); \\(L(\\Lambda)\\) is defined in the same way. A \\({\\rm II}_1\\) factor is an infinite-dimensional von Neumann algebra with center \\(\\mathbb C I\\) and a faithful normal tracial state.",
    "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThe groups are not isomorphic as abstract groups:\n\\[\n\\Gamma_1\\not\\cong\\Gamma_2.\n\\]",
    "target_statement": "A discrete group has property \\((T)\\) if every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), there is a unit vector \\(\\xi\\) with \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\nA group is ICC if the conjugacy class of every nonidentity element is infinite.\n\nThere exist two countable ICC groups \\(\\Gamma\\) and \\(\\Lambda\\), both with property \\((T)\\), which are not isomorphic as abstract groups, but whose group von Neumann algebras are trace-preservingly normally \\(*\\)-isomorphic \\({\\rm II}_1\\) factors:\n\\[\n\\Gamma\\not\\cong\\Lambda,\\qquad L(\\Gamma)\\cong L(\\Lambda).\n\\]\nThe algebra \\(L(\\Gamma)\\) is generated by the left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\); \\(L(\\Lambda)\\) is defined in the same way. A \\({\\rm II}_1\\) factor is an infinite-dimensional von Neumann algebra with center \\(\\mathbb C I\\) and a faithful normal tracial state.",
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  "SM-1-H0-to-H1": {
    "id": "SM-1-H0-to-H1",
    "problem_id": "SM-1",
    "problem_name": "Jamming exponent identity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 1,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H1: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nThe function \\(f\\) extends to \\(C^2(\\mathbb R)\\) and satisfies\n\\[\n\\frac c2f''+\\left((b-cH_-)t-\\frac{c^2}{2}J'\\right)f'+cf(1-f)=0\n\\]\non the entire real line, where the coefficient at \\(t=0\\) is defined by its continuous limit.",
    "starting_statement": null,
    "target_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nThe function \\(f\\) extends to \\(C^2(\\mathbb R)\\) and satisfies\n\\[\n\\frac c2f''+\\left((b-cH_-)t-\\frac{c^2}{2}J'\\right)f'+cf(1-f)=0\n\\]\non the entire real line, where the coefficient at \\(t=0\\) is defined by its continuous limit.",
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  "SM-1-H0-to-H2": {
    "id": "SM-1-H0-to-H2",
    "problem_id": "SM-1",
    "problem_name": "Jamming exponent identity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 2,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H2: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nThe following improper integral, with zero omitted from its integrand, is absolutely convergent and equals zero:\n\\[\n\\int_{\\mathbb R}p\\left\\{c\\left[(f')^2-2f^2(1-f)\\right]-(b-a-c)f(1-f)\\right\\}\\,dt=0.\n\\]",
    "starting_statement": null,
    "target_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nThe following improper integral, with zero omitted from its integrand, is absolutely convergent and equals zero:\n\\[\n\\int_{\\mathbb R}p\\left\\{c\\left[(f')^2-2f^2(1-f)\\right]-(b-a-c)f(1-f)\\right\\}\\,dt=0.\n\\]",
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  "SM-1-H0-to-H3": {
    "id": "SM-1-H0-to-H3",
    "problem_id": "SM-1",
    "problem_name": "Jamming exponent identity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H3: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(p:\\mathbb R\\to(0,\\infty)\\) be continuous and grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nThen\n\\[\n\\int_{\\mathbb R}p(f')^2\\,dt=2\\int_{\\mathbb R}pf^2(1-f)\\,dt>0.\n\\]",
    "starting_statement": null,
    "target_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(p:\\mathbb R\\to(0,\\infty)\\) be continuous and grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nThen\n\\[\n\\int_{\\mathbb R}p(f')^2\\,dt=2\\int_{\\mathbb R}pf^2(1-f)\\,dt>0.\n\\]",
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  "SM-1-H0-to-H4": {
    "id": "SM-1-H0-to-H4",
    "problem_id": "SM-1",
    "problem_name": "Jamming exponent identity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H4: Let \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nThen \\(0\\le F(\\tau,h)\\le1\\) for every \\(\\tau\\ge0\\) and \\(h\\in\\mathbb R\\).",
    "starting_statement": null,
    "target_statement": "Let \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nThen \\(0\\le F(\\tau,h)\\le1\\) for every \\(\\tau\\ge0\\) and \\(h\\in\\mathbb R\\).",
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  "SM-1-H0-to-H5": {
    "id": "SM-1-H0-to-H5",
    "problem_id": "SM-1",
    "problem_name": "Jamming exponent identity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove H5: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nLet \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nSuppose there are \\(\\tau_n\\ge0\\) and maps \\(h_n:\\mathbb R\\setminus\\{0\\}\\to\\mathbb R\\) with\n\\[\nF(\\tau_n,h_n(t))\\longrightarrow f(t)\\qquad(t\\ne0).\n\\]\nThen\n\\[\na+b=1.\n\\]",
    "starting_statement": null,
    "target_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nLet \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nSuppose there are \\(\\tau_n\\ge0\\) and maps \\(h_n:\\mathbb R\\setminus\\{0\\}\\to\\mathbb R\\) with\n\\[\nF(\\tau_n,h_n(t))\\longrightarrow f(t)\\qquad(t\\ne0).\n\\]\nThen\n\\[\na+b=1.\n\\]",
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  "SM-1-H1-to-H2": {
    "id": "SM-1-H1-to-H2",
    "problem_id": "SM-1",
    "problem_name": "Jamming exponent identity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 2,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nThe function \\(f\\) extends to \\(C^2(\\mathbb R)\\) and satisfies\n\\[\n\\frac c2f''+\\left((b-cH_-)t-\\frac{c^2}{2}J'\\right)f'+cf(1-f)=0\n\\]\non the entire real line, where the coefficient at \\(t=0\\) is defined by its continuous limit.\n\nProve H2: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nThe following improper integral, with zero omitted from its integrand, is absolutely convergent and equals zero:\n\\[\n\\int_{\\mathbb R}p\\left\\{c\\left[(f')^2-2f^2(1-f)\\right]-(b-a-c)f(1-f)\\right\\}\\,dt=0.\n\\]",
    "starting_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nThe function \\(f\\) extends to \\(C^2(\\mathbb R)\\) and satisfies\n\\[\n\\frac c2f''+\\left((b-cH_-)t-\\frac{c^2}{2}J'\\right)f'+cf(1-f)=0\n\\]\non the entire real line, where the coefficient at \\(t=0\\) is defined by its continuous limit.",
    "target_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nThe following improper integral, with zero omitted from its integrand, is absolutely convergent and equals zero:\n\\[\n\\int_{\\mathbb R}p\\left\\{c\\left[(f')^2-2f^2(1-f)\\right]-(b-a-c)f(1-f)\\right\\}\\,dt=0.\n\\]",
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  "SM-1-H1-to-H3": {
    "id": "SM-1-H1-to-H3",
    "problem_id": "SM-1",
    "problem_name": "Jamming exponent identity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nThe function \\(f\\) extends to \\(C^2(\\mathbb R)\\) and satisfies\n\\[\n\\frac c2f''+\\left((b-cH_-)t-\\frac{c^2}{2}J'\\right)f'+cf(1-f)=0\n\\]\non the entire real line, where the coefficient at \\(t=0\\) is defined by its continuous limit.\n\nProve H3: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(p:\\mathbb R\\to(0,\\infty)\\) be continuous and grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nThen\n\\[\n\\int_{\\mathbb R}p(f')^2\\,dt=2\\int_{\\mathbb R}pf^2(1-f)\\,dt>0.\n\\]",
    "starting_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nThe function \\(f\\) extends to \\(C^2(\\mathbb R)\\) and satisfies\n\\[\n\\frac c2f''+\\left((b-cH_-)t-\\frac{c^2}{2}J'\\right)f'+cf(1-f)=0\n\\]\non the entire real line, where the coefficient at \\(t=0\\) is defined by its continuous limit.",
    "target_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(p:\\mathbb R\\to(0,\\infty)\\) be continuous and grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nThen\n\\[\n\\int_{\\mathbb R}p(f')^2\\,dt=2\\int_{\\mathbb R}pf^2(1-f)\\,dt>0.\n\\]",
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  "SM-1-H1-to-H4": {
    "id": "SM-1-H1-to-H4",
    "problem_id": "SM-1",
    "problem_name": "Jamming exponent identity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nThe function \\(f\\) extends to \\(C^2(\\mathbb R)\\) and satisfies\n\\[\n\\frac c2f''+\\left((b-cH_-)t-\\frac{c^2}{2}J'\\right)f'+cf(1-f)=0\n\\]\non the entire real line, where the coefficient at \\(t=0\\) is defined by its continuous limit.\n\nProve H4: Let \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nThen \\(0\\le F(\\tau,h)\\le1\\) for every \\(\\tau\\ge0\\) and \\(h\\in\\mathbb R\\).",
    "starting_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nThe function \\(f\\) extends to \\(C^2(\\mathbb R)\\) and satisfies\n\\[\n\\frac c2f''+\\left((b-cH_-)t-\\frac{c^2}{2}J'\\right)f'+cf(1-f)=0\n\\]\non the entire real line, where the coefficient at \\(t=0\\) is defined by its continuous limit.",
    "target_statement": "Let \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nThen \\(0\\le F(\\tau,h)\\le1\\) for every \\(\\tau\\ge0\\) and \\(h\\in\\mathbb R\\).",
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  "SM-1-H1-to-H5": {
    "id": "SM-1-H1-to-H5",
    "problem_id": "SM-1",
    "problem_name": "Jamming exponent identity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H1: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nThe function \\(f\\) extends to \\(C^2(\\mathbb R)\\) and satisfies\n\\[\n\\frac c2f''+\\left((b-cH_-)t-\\frac{c^2}{2}J'\\right)f'+cf(1-f)=0\n\\]\non the entire real line, where the coefficient at \\(t=0\\) is defined by its continuous limit.\n\nProve H5: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nLet \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nSuppose there are \\(\\tau_n\\ge0\\) and maps \\(h_n:\\mathbb R\\setminus\\{0\\}\\to\\mathbb R\\) with\n\\[\nF(\\tau_n,h_n(t))\\longrightarrow f(t)\\qquad(t\\ne0).\n\\]\nThen\n\\[\na+b=1.\n\\]",
    "starting_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nThe function \\(f\\) extends to \\(C^2(\\mathbb R)\\) and satisfies\n\\[\n\\frac c2f''+\\left((b-cH_-)t-\\frac{c^2}{2}J'\\right)f'+cf(1-f)=0\n\\]\non the entire real line, where the coefficient at \\(t=0\\) is defined by its continuous limit.",
    "target_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nLet \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nSuppose there are \\(\\tau_n\\ge0\\) and maps \\(h_n:\\mathbb R\\setminus\\{0\\}\\to\\mathbb R\\) with\n\\[\nF(\\tau_n,h_n(t))\\longrightarrow f(t)\\qquad(t\\ne0).\n\\]\nThen\n\\[\na+b=1.\n\\]",
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  "SM-1-H2-to-H3": {
    "id": "SM-1-H2-to-H3",
    "problem_id": "SM-1",
    "problem_name": "Jamming exponent identity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nThe following improper integral, with zero omitted from its integrand, is absolutely convergent and equals zero:\n\\[\n\\int_{\\mathbb R}p\\left\\{c\\left[(f')^2-2f^2(1-f)\\right]-(b-a-c)f(1-f)\\right\\}\\,dt=0.\n\\]\n\nProve H3: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(p:\\mathbb R\\to(0,\\infty)\\) be continuous and grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nThen\n\\[\n\\int_{\\mathbb R}p(f')^2\\,dt=2\\int_{\\mathbb R}pf^2(1-f)\\,dt>0.\n\\]",
    "starting_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nThe following improper integral, with zero omitted from its integrand, is absolutely convergent and equals zero:\n\\[\n\\int_{\\mathbb R}p\\left\\{c\\left[(f')^2-2f^2(1-f)\\right]-(b-a-c)f(1-f)\\right\\}\\,dt=0.\n\\]",
    "target_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(p:\\mathbb R\\to(0,\\infty)\\) be continuous and grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nThen\n\\[\n\\int_{\\mathbb R}p(f')^2\\,dt=2\\int_{\\mathbb R}pf^2(1-f)\\,dt>0.\n\\]",
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  "SM-1-H2-to-H4": {
    "id": "SM-1-H2-to-H4",
    "problem_id": "SM-1",
    "problem_name": "Jamming exponent identity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nThe following improper integral, with zero omitted from its integrand, is absolutely convergent and equals zero:\n\\[\n\\int_{\\mathbb R}p\\left\\{c\\left[(f')^2-2f^2(1-f)\\right]-(b-a-c)f(1-f)\\right\\}\\,dt=0.\n\\]\n\nProve H4: Let \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nThen \\(0\\le F(\\tau,h)\\le1\\) for every \\(\\tau\\ge0\\) and \\(h\\in\\mathbb R\\).",
    "starting_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nThe following improper integral, with zero omitted from its integrand, is absolutely convergent and equals zero:\n\\[\n\\int_{\\mathbb R}p\\left\\{c\\left[(f')^2-2f^2(1-f)\\right]-(b-a-c)f(1-f)\\right\\}\\,dt=0.\n\\]",
    "target_statement": "Let \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nThen \\(0\\le F(\\tau,h)\\le1\\) for every \\(\\tau\\ge0\\) and \\(h\\in\\mathbb R\\).",
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  "SM-1-H2-to-H5": {
    "id": "SM-1-H2-to-H5",
    "problem_id": "SM-1",
    "problem_name": "Jamming exponent identity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H2: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nThe following improper integral, with zero omitted from its integrand, is absolutely convergent and equals zero:\n\\[\n\\int_{\\mathbb R}p\\left\\{c\\left[(f')^2-2f^2(1-f)\\right]-(b-a-c)f(1-f)\\right\\}\\,dt=0.\n\\]\n\nProve H5: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nLet \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nSuppose there are \\(\\tau_n\\ge0\\) and maps \\(h_n:\\mathbb R\\setminus\\{0\\}\\to\\mathbb R\\) with\n\\[\nF(\\tau_n,h_n(t))\\longrightarrow f(t)\\qquad(t\\ne0).\n\\]\nThen\n\\[\na+b=1.\n\\]",
    "starting_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nThe following improper integral, with zero omitted from its integrand, is absolutely convergent and equals zero:\n\\[\n\\int_{\\mathbb R}p\\left\\{c\\left[(f')^2-2f^2(1-f)\\right]-(b-a-c)f(1-f)\\right\\}\\,dt=0.\n\\]",
    "target_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nLet \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nSuppose there are \\(\\tau_n\\ge0\\) and maps \\(h_n:\\mathbb R\\setminus\\{0\\}\\to\\mathbb R\\) with\n\\[\nF(\\tau_n,h_n(t))\\longrightarrow f(t)\\qquad(t\\ne0).\n\\]\nThen\n\\[\na+b=1.\n\\]",
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  "SM-1-H3-to-H4": {
    "id": "SM-1-H3-to-H4",
    "problem_id": "SM-1",
    "problem_name": "Jamming exponent identity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H3: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(p:\\mathbb R\\to(0,\\infty)\\) be continuous and grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nThen\n\\[\n\\int_{\\mathbb R}p(f')^2\\,dt=2\\int_{\\mathbb R}pf^2(1-f)\\,dt>0.\n\\]\n\nProve H4: Let \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nThen \\(0\\le F(\\tau,h)\\le1\\) for every \\(\\tau\\ge0\\) and \\(h\\in\\mathbb R\\).",
    "starting_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(p:\\mathbb R\\to(0,\\infty)\\) be continuous and grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nThen\n\\[\n\\int_{\\mathbb R}p(f')^2\\,dt=2\\int_{\\mathbb R}pf^2(1-f)\\,dt>0.\n\\]",
    "target_statement": "Let \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nThen \\(0\\le F(\\tau,h)\\le1\\) for every \\(\\tau\\ge0\\) and \\(h\\in\\mathbb R\\).",
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  "SM-1-H3-to-H5": {
    "id": "SM-1-H3-to-H5",
    "problem_id": "SM-1",
    "problem_name": "Jamming exponent identity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H3: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(p:\\mathbb R\\to(0,\\infty)\\) be continuous and grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nThen\n\\[\n\\int_{\\mathbb R}p(f')^2\\,dt=2\\int_{\\mathbb R}pf^2(1-f)\\,dt>0.\n\\]\n\nProve H5: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nLet \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nSuppose there are \\(\\tau_n\\ge0\\) and maps \\(h_n:\\mathbb R\\setminus\\{0\\}\\to\\mathbb R\\) with\n\\[\nF(\\tau_n,h_n(t))\\longrightarrow f(t)\\qquad(t\\ne0).\n\\]\nThen\n\\[\na+b=1.\n\\]",
    "starting_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(p:\\mathbb R\\to(0,\\infty)\\) be continuous and grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nThen\n\\[\n\\int_{\\mathbb R}p(f')^2\\,dt=2\\int_{\\mathbb R}pf^2(1-f)\\,dt>0.\n\\]",
    "target_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nLet \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nSuppose there are \\(\\tau_n\\ge0\\) and maps \\(h_n:\\mathbb R\\setminus\\{0\\}\\to\\mathbb R\\) with\n\\[\nF(\\tau_n,h_n(t))\\longrightarrow f(t)\\qquad(t\\ne0).\n\\]\nThen\n\\[\na+b=1.\n\\]",
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  "SM-1-H4-to-H5": {
    "id": "SM-1-H4-to-H5",
    "problem_id": "SM-1",
    "problem_name": "Jamming exponent identity",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 4,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "H4: Let \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nThen \\(0\\le F(\\tau,h)\\le1\\) for every \\(\\tau\\ge0\\) and \\(h\\in\\mathbb R\\).\n\nProve H5: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nLet \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nSuppose there are \\(\\tau_n\\ge0\\) and maps \\(h_n:\\mathbb R\\setminus\\{0\\}\\to\\mathbb R\\) with\n\\[\nF(\\tau_n,h_n(t))\\longrightarrow f(t)\\qquad(t\\ne0).\n\\]\nThen\n\\[\na+b=1.\n\\]",
    "starting_statement": "Let \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nThen \\(0\\le F(\\tau,h)\\le1\\) for every \\(\\tau\\ge0\\) and \\(h\\in\\mathbb R\\).",
    "target_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nLet \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nSuppose there are \\(\\tau_n\\ge0\\) and maps \\(h_n:\\mathbb R\\setminus\\{0\\}\\to\\mathbb R\\) with\n\\[\nF(\\tau_n,h_n(t))\\longrightarrow f(t)\\qquad(t\\ne0).\n\\]\nThen\n\\[\na+b=1.\n\\]",
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  "SP-1-H0-to-H1": {
    "id": "SP-1-H0-to-H1",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 1,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "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\\]",
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  "SP-1-H0-to-H2": {
    "id": "SP-1-H0-to-H2",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 2,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove 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": 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 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)]\\).",
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  "SP-1-H0-to-H3": {
    "id": "SP-1-H0-to-H3",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove 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.",
    "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 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.",
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  "SP-1-H0-to-H4": {
    "id": "SP-1-H0-to-H4",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 0,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "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\\]",
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  "SP-1-H0-to-H5": {
    "id": "SP-1-H0-to-H5",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
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    "start": 0,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "Prove 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": 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, 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\\]",
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  "SP-1-H0-to-H6": {
    "id": "SP-1-H0-to-H6",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
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    "start": 0,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.",
    "user": "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.",
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  "SP-1-H1-to-H2": {
    "id": "SP-1-H1-to-H2",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 2,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "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)]\\).",
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  "SP-1-H1-to-H3": {
    "id": "SP-1-H1-to-H3",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "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 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.",
    "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 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.",
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  "SP-1-H1-to-H4": {
    "id": "SP-1-H1-to-H4",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "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 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": "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 \\(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\\]",
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  "SP-1-H1-to-H5": {
    "id": "SP-1-H1-to-H5",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "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 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": "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": "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\\]",
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  "SP-1-H1-to-H6": {
    "id": "SP-1-H1-to-H6",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 1,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "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 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": "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": "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.",
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  "SP-1-H2-to-H3": {
    "id": "SP-1-H2-to-H3",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 3,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "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)]\\).\n\nProve 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.",
    "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 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)]\\).",
    "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 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.",
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  "SP-1-H2-to-H4": {
    "id": "SP-1-H2-to-H4",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "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)]\\).\n\nProve 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": "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)]\\).",
    "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\\]",
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  "SP-1-H2-to-H5": {
    "id": "SP-1-H2-to-H5",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "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)]\\).\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 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)]\\).",
    "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\\]",
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  "SP-1-H2-to-H6": {
    "id": "SP-1-H2-to-H6",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 2,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "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)]\\).\n\nProve 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": "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)]\\).",
    "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.",
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  "SP-1-H3-to-H4": {
    "id": "SP-1-H3-to-H4",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 4,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "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 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": "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": "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\\]",
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  "SP-1-H3-to-H5": {
    "id": "SP-1-H3-to-H5",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "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\\]",
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  "SP-1-H3-to-H6": {
    "id": "SP-1-H3-to-H6",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 3,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "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 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": "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\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.",
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  "SP-1-H4-to-H5": {
    "id": "SP-1-H4-to-H5",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
    "start": 4,
    "target": 5,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "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\\]\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 \\(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\\]",
    "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\\]",
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  "SP-1-H4-to-H6": {
    "id": "SP-1-H4-to-H6",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
    "eval_version": "other-v5",
    "protocol_id": "bare-proof-other-v5",
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    "target": 6,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "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\\]\n\nProve 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": "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\\]",
    "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.",
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  "SP-1-H5-to-H6": {
    "id": "SP-1-H5-to-H6",
    "problem_id": "SP-1",
    "problem_name": "Quartic SYK spectral edge",
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    "start": 5,
    "target": 6,
    "system": "You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.",
    "user": "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\\]\n\nProve 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": "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\\]",
    "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.",
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