{
  "EP-1-T01": {
    "id": "EP-1-T01",
    "problem_id": "EP-1",
    "label": "H1",
    "title": "Coefficient algebra and Dirichlet operators",
    "setup": "",
    "target": "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\\]",
    "permitted_results": [],
    "case_id": "EP-1-H0-to-H1",
    "eval_version": "other-v5",
    "mathematically_validated": false,
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  "EP-1-T02": {
    "id": "EP-1-T02",
    "problem_id": "EP-1",
    "label": "H2",
    "title": "Conformal reconstruction",
    "setup": "",
    "target": "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.",
    "permitted_results": [],
    "case_id": "EP-1-H0-to-H2",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "EP-1-T03": {
    "id": "EP-1-T03",
    "problem_id": "EP-1",
    "label": "H3",
    "title": "Residual and derivative bounds",
    "setup": "",
    "target": "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\\]",
    "permitted_results": [],
    "case_id": "EP-1-H0-to-H3",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "EP-1-T04": {
    "id": "EP-1-T04",
    "problem_id": "EP-1",
    "label": "H4",
    "title": "Nonconstant solution on the disk",
    "setup": "",
    "target": "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.",
    "permitted_results": [],
    "case_id": "EP-1-H0-to-H4",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "EP-1-T05": {
    "id": "EP-1-T05",
    "problem_id": "EP-1",
    "label": "H5",
    "title": "Planar Berenstein counterexample",
    "setup": "",
    "target": "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.",
    "permitted_results": [],
    "case_id": "EP-1-H0-to-H5",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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    }
  },
  "KT-1-T01": {
    "id": "KT-1-T01",
    "problem_id": "KT-1",
    "label": "H1",
    "title": "Odd-linking polynomial splitting",
    "setup": "",
    "target": "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}]\\).",
    "permitted_results": [],
    "case_id": "KT-1-H0-to-H1",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "KT-1-T02": {
    "id": "KT-1-T02",
    "problem_id": "KT-1",
    "label": "H2",
    "title": "Involutive symmetry and the Conway square",
    "setup": "",
    "target": "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\\).",
    "permitted_results": [],
    "case_id": "KT-1-H0-to-H2",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "KT-1-T03": {
    "id": "KT-1-T03",
    "problem_id": "KT-1",
    "label": "H3",
    "title": "Double-cover Alexander factorization",
    "setup": "",
    "target": "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\\).",
    "permitted_results": [],
    "case_id": "KT-1-H0-to-H3",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "KT-1-T04": {
    "id": "KT-1-T04",
    "problem_id": "KT-1",
    "label": "H4",
    "title": "Integral Alexander normalization",
    "setup": "",
    "target": "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\\).",
    "permitted_results": [],
    "case_id": "KT-1-H0-to-H4",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  "KT-1-T05": {
    "id": "KT-1-T05",
    "problem_id": "KT-1",
    "label": "H5",
    "title": "Finite-symmetry modulo-four splitting",
    "setup": "",
    "target": "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.",
    "permitted_results": [],
    "case_id": "KT-1-H0-to-H5",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "KT-1-T06": {
    "id": "KT-1-T06",
    "problem_id": "KT-1",
    "label": "H6",
    "title": "Amphicheiral modulo-four splitting",
    "setup": "",
    "target": "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.",
    "permitted_results": [],
    "case_id": "KT-1-H0-to-H6",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "NE-1-T01": {
    "id": "NE-1-T01",
    "problem_id": "NE-1",
    "label": "H1",
    "title": "Propagation of the observation",
    "setup": "",
    "target": "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\\]",
    "permitted_results": [],
    "case_id": "NE-1-H0-to-H1",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "NE-1-T02": {
    "id": "NE-1-T02",
    "problem_id": "NE-1",
    "label": "H2",
    "title": "Weighted evolution identity",
    "setup": "",
    "target": "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\\]",
    "permitted_results": [],
    "case_id": "NE-1-H0-to-H2",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "NE-1-T03": {
    "id": "NE-1-T03",
    "problem_id": "NE-1",
    "label": "H3",
    "title": "Finite total flux",
    "setup": "",
    "target": "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\\]",
    "permitted_results": [],
    "case_id": "NE-1-H0-to-H3",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
      "checkpoints_sha256": "996068675b8b3f9fb6a899bac096a7901d57e5d9faa81e9ebaf4ae52c38feeaa",
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    }
  },
  "NE-1-T04": {
    "id": "NE-1-T04",
    "problem_id": "NE-1",
    "label": "H4",
    "title": "Global rigidity",
    "setup": "",
    "target": "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\\]",
    "permitted_results": [],
    "case_id": "NE-1-H0-to-H4",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
      "checkpoints_sha256": "996068675b8b3f9fb6a899bac096a7901d57e5d9faa81e9ebaf4ae52c38feeaa",
      "protocol_sha256": "12638b90b4a8149985609c1890c1f7d6c77c519b7b526f78c98ad20e6d2149fe"
    }
  },
  "OA-1-T01": {
    "id": "OA-1-T01",
    "problem_id": "OA-1",
    "label": "H1",
    "title": "Completion matrix inequality",
    "setup": "",
    "target": "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^*\\).",
    "permitted_results": [],
    "case_id": "OA-1-H0-to-H1",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
      "checkpoints_sha256": "996068675b8b3f9fb6a899bac096a7901d57e5d9faa81e9ebaf4ae52c38feeaa",
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    }
  },
  "OA-1-T02": {
    "id": "OA-1-T02",
    "problem_id": "OA-1",
    "label": "H2",
    "title": "Completion norm bound",
    "setup": "",
    "target": "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.",
    "permitted_results": [],
    "case_id": "OA-1-H0-to-H2",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
      "checkpoints_sha256": "996068675b8b3f9fb6a899bac096a7901d57e5d9faa81e9ebaf4ae52c38feeaa",
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    }
  },
  "OA-1-T03": {
    "id": "OA-1-T03",
    "problem_id": "OA-1",
    "label": "H3",
    "title": "Positive unital boundary map",
    "setup": "",
    "target": "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\\).",
    "permitted_results": [],
    "case_id": "OA-1-H0-to-H3",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
      "checkpoints_sha256": "996068675b8b3f9fb6a899bac096a7901d57e5d9faa81e9ebaf4ae52c38feeaa",
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  },
  "OA-1-T04": {
    "id": "OA-1-T04",
    "problem_id": "OA-1",
    "label": "H4",
    "title": "Simple-spectrum numerical-range bound",
    "setup": "",
    "target": "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.",
    "permitted_results": [],
    "case_id": "OA-1-H0-to-H4",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
      "checkpoints_sha256": "996068675b8b3f9fb6a899bac096a7901d57e5d9faa81e9ebaf4ae52c38feeaa",
      "protocol_sha256": "12638b90b4a8149985609c1890c1f7d6c77c519b7b526f78c98ad20e6d2149fe"
    }
  },
  "OA-1-T05": {
    "id": "OA-1-T05",
    "problem_id": "OA-1",
    "label": "H5",
    "title": "Scalar numerical-range inequality",
    "setup": "",
    "target": "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\\).",
    "permitted_results": [],
    "case_id": "OA-1-H0-to-H5",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
      "checkpoints_sha256": "996068675b8b3f9fb6a899bac096a7901d57e5d9faa81e9ebaf4ae52c38feeaa",
      "protocol_sha256": "12638b90b4a8149985609c1890c1f7d6c77c519b7b526f78c98ad20e6d2149fe"
    }
  },
  "OA-2-T01": {
    "id": "OA-2-T01",
    "problem_id": "OA-2",
    "label": "H1",
    "title": "Semidirect-product realization",
    "setup": "",
    "target": "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\\]",
    "permitted_results": [],
    "case_id": "OA-2-H0-to-H1",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
      "checkpoints_sha256": "996068675b8b3f9fb6a899bac096a7901d57e5d9faa81e9ebaf4ae52c38feeaa",
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  },
  "OA-2-T02": {
    "id": "OA-2-T02",
    "problem_id": "OA-2",
    "label": "H2",
    "title": "Group von Neumann algebra isomorphism",
    "setup": "",
    "target": "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}\\).",
    "permitted_results": [],
    "case_id": "OA-2-H0-to-H2",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "OA-2-T03": {
    "id": "OA-2-T03",
    "problem_id": "OA-2",
    "label": "H3",
    "title": "Uniform invariant-measure detection",
    "setup": "",
    "target": "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\\]",
    "permitted_results": [],
    "case_id": "OA-2-H0-to-H3",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "OA-2-T04": {
    "id": "OA-2-T04",
    "problem_id": "OA-2",
    "label": "H4",
    "title": "Property (T)",
    "setup": "",
    "target": "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\\).",
    "permitted_results": [],
    "case_id": "OA-2-H0-to-H4",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "OA-2-T05": {
    "id": "OA-2-T05",
    "problem_id": "OA-2",
    "label": "H5",
    "title": "Abstract group nonisomorphism",
    "setup": "",
    "target": "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\\]",
    "permitted_results": [],
    "case_id": "OA-2-H0-to-H5",
    "eval_version": "other-v5",
    "mathematically_validated": false,
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  "OA-2-T06": {
    "id": "OA-2-T06",
    "problem_id": "OA-2",
    "label": "H6",
    "title": "Nonisomorphic groups with isomorphic factors",
    "setup": "",
    "target": "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.",
    "permitted_results": [],
    "case_id": "OA-2-H0-to-H6",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "SM-1-T01": {
    "id": "SM-1-T01",
    "problem_id": "SM-1",
    "label": "H1",
    "title": "Classical profile equation",
    "setup": "",
    "target": "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.",
    "permitted_results": [],
    "case_id": "SM-1-H0-to-H1",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "SM-1-T02": {
    "id": "SM-1-T02",
    "problem_id": "SM-1",
    "label": "H2",
    "title": "Integrated exponent defect",
    "setup": "",
    "target": "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\\]",
    "permitted_results": [],
    "case_id": "SM-1-H0-to-H2",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "SM-1-T03": {
    "id": "SM-1-T03",
    "problem_id": "SM-1",
    "label": "H3",
    "title": "Integrated marginality",
    "setup": "",
    "target": "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\\]",
    "permitted_results": [],
    "case_id": "SM-1-H0-to-H3",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "SM-1-T04": {
    "id": "SM-1-T04",
    "problem_id": "SM-1",
    "label": "H4",
    "title": "Parabolic range preservation",
    "setup": "",
    "target": "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\\).",
    "permitted_results": [],
    "case_id": "SM-1-H0-to-H4",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "SM-1-T05": {
    "id": "SM-1-T05",
    "problem_id": "SM-1",
    "label": "H5",
    "title": "Conditional jamming exponent identity",
    "setup": "",
    "target": "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\\]",
    "permitted_results": [],
    "case_id": "SM-1-H0-to-H5",
    "eval_version": "other-v5",
    "mathematically_validated": false,
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  "SP-1-T01": {
    "id": "SP-1-T01",
    "problem_id": "SP-1",
    "label": "H1",
    "title": "Positive thermal fixed point",
    "setup": "",
    "target": "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\\]",
    "permitted_results": [],
    "case_id": "SP-1-H0-to-H1",
    "eval_version": "other-v5",
    "mathematically_validated": false,
    "source_hashes": {
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  },
  "SP-1-T02": {
    "id": "SP-1-T02",
    "problem_id": "SP-1",
    "label": "H2",
    "title": "History-weighted locality",
    "setup": "",
    "target": "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)]\\).",
    "permitted_results": [],
    "case_id": "SP-1-H0-to-H2",
    "eval_version": "other-v5",
    "mathematically_validated": false,
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  "SP-1-T03": {
    "id": "SP-1-T03",
    "problem_id": "SP-1",
    "label": "H3",
    "title": "Common cavity directing law",
    "setup": "",
    "target": "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.",
    "permitted_results": [],
    "case_id": "SP-1-H0-to-H3",
    "eval_version": "other-v5",
    "mathematically_validated": false,
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  "SP-1-T04": {
    "id": "SP-1-T04",
    "problem_id": "SP-1",
    "label": "H4",
    "title": "Finite-bath thermal kernel convergence",
    "setup": "",
    "target": "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\\]",
    "permitted_results": [],
    "case_id": "SP-1-H0-to-H4",
    "eval_version": "other-v5",
    "mathematically_validated": false,
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  "SP-1-T05": {
    "id": "SP-1-T05",
    "problem_id": "SP-1",
    "label": "H5",
    "title": "Annealed pressure limit",
    "setup": "",
    "target": "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\\]",
    "permitted_results": [],
    "case_id": "SP-1-H0-to-H5",
    "eval_version": "other-v5",
    "mathematically_validated": false,
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  "SP-1-T06": {
    "id": "SP-1-T06",
    "problem_id": "SP-1",
    "label": "H6",
    "title": "Almost-sure spectral edge",
    "setup": "",
    "target": "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.",
    "permitted_results": [],
    "case_id": "SP-1-H0-to-H6",
    "eval_version": "other-v5",
    "mathematically_validated": false,
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