{ "schema_version": "dynacertbench.executable", "description": "Question-only executable DynaCertBench ordered by active LaTeX systeminstance blocks.", "source_dataset": "dynacertbench_by_problem.json", "paper_snapshot_id": "paper", "record_count": 105, "excluded_count": 0, "records": [ { "record_id": "DCB-LIT-P1-001", "problem": { "problem_id": "dcb_lit_p1_001", "task_type": "stability", "name": "DynaCertBench System 001 · DCB-LIT-P1-001 · Exercise~4.30, \\cite{khalil2002nonlinear}", "dynamics": "dx1/dt = -((1)/(2))*tan(((pi*x1)/(2))+((pi)/(4)))+x2+((1)/(2)), dx2/dt = x1-((1)/(2))*tan(((pi*x2)/(2))+((pi)/(4)))+((1)/(2))", "source": "paper System 1; canonical record DCB-LIT-P1-001", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-001.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner radius is provenance only; formal N includes the equilibrium." ], "equilibrium": [ 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "constraints", "constraints": [ { "lhs": "x1**2 + x2**2", "relation": "le", "rhs": "(0.25)**2" } ], "bounds": [ [ "-0.25", "0.25" ], [ "-0.25", "0.25" ] ] }, "analysis_bounds": [ [ "-0.25", "0.25" ], [ "-0.25", "0.25" ] ] }, "candidate": null, "paper_system_number": 1, "paper_system_label": "System 1", "local_name": "S001__DCB-LIT-P1-001" }, { "record_id": "DCB-LIT-P1-002", "problem": { "problem_id": "dcb_lit_p1_002", "task_type": "stability", "name": "DynaCertBench System 002 · DCB-LIT-P1-002 · Table~1 row~1 (NonPoly0), \\cite{abate2021fossil}", "dynamics": "dx1/dt = -x1+x1*x2, dx2/dt = -x2", "source": "paper System 2; canonical record DCB-LIT-P1-002", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-002.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner radius is provenance only; formal N includes the equilibrium." ], "equilibrium": [ 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "constraints", "constraints": [ { "lhs": "x1**2 + x2**2", "relation": "le", "rhs": "(1)**2" } ], "bounds": [ [ "-1", "1" ], [ "-1", "1" ] ] }, "analysis_bounds": [ [ "-1", "1" ], [ "-1", "1" ] ] }, "candidate": null, "paper_system_number": 2, "paper_system_label": "System 2", "local_name": "S002__DCB-LIT-P1-002" }, { "record_id": "DCB-LIT-P1-003", "problem": { "problem_id": "dcb_lit_p1_003", "task_type": "stability", "name": "DynaCertBench System 003 · DCB-LIT-P1-003 · Table~1 row~2 (NonPoly1), \\cite{abate2021fossil}", "dynamics": "dx1/dt = -x1+2*x1**2*x2, dx2/dt = -x2", "source": "paper System 3; canonical record DCB-LIT-P1-003", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-003.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "equilibrium": [ 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "constraints", "constraints": [ { "lhs": "x1**2 + x2**2", "relation": "le", "rhs": "(10)**2" }, { "lhs": "x1", "relation": "ge", "rhs": "0" }, { "lhs": "x2", "relation": "ge", "rhs": "0" } ], "bounds": [ [ "0", "10" ], [ "0", "10" ] ] }, "analysis_bounds": [ [ "0", "10" ], [ "0", "10" ] ] }, "candidate": null, "paper_system_number": 3, "paper_system_label": "System 3", "local_name": "S003__DCB-LIT-P1-003" }, { "record_id": "DCB-LIT-P1-004", "problem": { "problem_id": "dcb_lit_p1_004", "task_type": "stability", "name": "DynaCertBench System 004 · DCB-LIT-P1-004 · Table~1 row~3 (NonPoly2), \\cite{abate2021fossil}", "dynamics": "dx1/dt = -x1, dx2/dt = -2*x2+0.1*x1*x2**2+x3, dx3/dt = -x3-1.5*x2", "source": "paper System 4; canonical record DCB-LIT-P1-004", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-004.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "equilibrium": [ 0, 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "constraints", "constraints": [ { "lhs": "x1**2 + x2**2 + x3**2", "relation": "le", "rhs": "(10)**2" }, { "lhs": "x1", "relation": "ge", "rhs": "0" }, { "lhs": "x2", "relation": "ge", "rhs": "0" }, { "lhs": "x3", "relation": "ge", "rhs": "0" } ], "bounds": [ [ "0", "10" ], [ "0", "10" ], [ "0", "10" ] ] }, "analysis_bounds": [ [ "0", "10" ], [ "0", "10" ], [ "0", "10" ] ] }, "candidate": null, "paper_system_number": 4, "paper_system_label": "System 4", "local_name": "S004__DCB-LIT-P1-004" }, { "record_id": "DCB-LIT-P1-005", "problem": { "problem_id": "dcb_lit_p1_005", "task_type": "stability", "name": "DynaCertBench System 005 · DCB-LIT-P1-005 · Table~1 row~4 (NonPoly3), \\cite{abate2021fossil}", "dynamics": "dx1/dt = -3*x1-0.1*x1*x2**3, dx2/dt = -x2+x3, dx3/dt = -x3", "source": "paper System 5; canonical record DCB-LIT-P1-005", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-005.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "equilibrium": [ 0, 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "constraints", "constraints": [ { "lhs": "x1**2 + x2**2 + x3**2", "relation": "le", "rhs": "(10)**2" }, { "lhs": "x1", "relation": "ge", "rhs": "0" }, { "lhs": "x2", "relation": "ge", "rhs": "0" }, { "lhs": "x3", "relation": "ge", "rhs": "0" } ], "bounds": [ [ "0", "10" ], [ "0", "10" ], [ "0", "10" ] ] }, "analysis_bounds": [ [ "0", "10" ], [ "0", "10" ], [ "0", "10" ] ] }, "candidate": null, "paper_system_number": 5, "paper_system_label": "System 5", "local_name": "S005__DCB-LIT-P1-005" }, { "record_id": "DCB-LIT-P1-006", "problem": { "problem_id": "dcb_lit_p1_006", "task_type": "stability", "name": "DynaCertBench System 006 · DCB-LIT-P1-006 · Table~1 row~5 (Poly1), \\cite{abate2021fossil}", "dynamics": "dx1/dt = -x1**3-x1*x3**2, dx2/dt = -x2-x1**2*x2, dx3/dt = -x3+3*x1**2*x3-3*x3", "source": "paper System 6; canonical record DCB-LIT-P1-006", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-006.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner radius is provenance only; formal N includes the equilibrium." ], "equilibrium": [ 0, 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "constraints", "constraints": [ { "lhs": "x1**2 + x2**2 + x3**2", "relation": "le", "rhs": "(10)**2" } ], "bounds": [ [ "-10", "10" ], [ "-10", "10" ], [ "-10", "10" ] ] }, "analysis_bounds": [ [ "-10", "10" ], [ "-10", "10" ], [ "-10", "10" ] ] }, "candidate": null, "paper_system_number": 6, "paper_system_label": "System 6", "local_name": "S006__DCB-LIT-P1-006" }, { "record_id": "DCB-LIT-P1-007", "problem": { "problem_id": "dcb_lit_p1_007", "task_type": "stability", "name": "DynaCertBench System 007 · DCB-LIT-P1-007 · Table~1 row~6 (Poly2), \\cite{abate2021fossil}", "dynamics": "dx1/dt = -x1**3+x2, dx2/dt = -x1-x2", "source": "paper System 7; canonical record DCB-LIT-P1-007", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-007.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner radius is provenance only; formal N includes the equilibrium." ], "equilibrium": [ 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "constraints", "constraints": [ { "lhs": "x1**2 + x2**2", "relation": "le", "rhs": "(10)**2" } ], "bounds": [ [ "-10", "10" ], [ "-10", "10" ] ] }, "analysis_bounds": [ [ "-10", "10" ], [ "-10", "10" ] ] }, "candidate": null, "paper_system_number": 7, "paper_system_label": "System 7", "local_name": "S007__DCB-LIT-P1-007" }, { "record_id": "DCB-LIT-P1-008", "problem": { "problem_id": "dcb_lit_p1_008", "task_type": "stability", "name": "DynaCertBench System 008 · DCB-LIT-P1-008 · Table~1 row~7 (Poly3), \\cite{abate2021fossil}", "dynamics": "dx1/dt = -x1**3-x2**2, dx2/dt = x1*x2-x2**3", "source": "paper System 8; canonical record DCB-LIT-P1-008", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-008.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner radius is provenance only; formal N includes the equilibrium." ], "equilibrium": [ 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "constraints", "constraints": [ { "lhs": "x1**2 + x2**2", "relation": "le", "rhs": "(10)**2" } ], "bounds": [ [ "-10", "10" ], [ "-10", "10" ] ] }, "analysis_bounds": [ [ "-10", "10" ], [ "-10", "10" ] ] }, "candidate": null, "paper_system_number": 8, "paper_system_label": "System 8", "local_name": "S008__DCB-LIT-P1-008" }, { "record_id": "DCB-LIT-P1-009", "problem": { "problem_id": "dcb_lit_p1_009", "task_type": "stability", "name": "DynaCertBench System 009 · DCB-LIT-P1-009 · Table~1 row~8 (Poly4), \\cite{abate2021fossil}", "dynamics": "dx1/dt = -x1-1.5*x1**2*x2**3, dx2/dt = -x2**3+0.5*x1**3*x2**2", "source": "paper System 9; canonical record DCB-LIT-P1-009", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-009.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner radius is provenance only; formal N includes the equilibrium." ], "equilibrium": [ 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "constraints", "constraints": [ { "lhs": "x1**2 + x2**2", "relation": "le", "rhs": "(10)**2" } ], "bounds": [ [ "-10", "10" ], [ "-10", "10" ] ] }, "analysis_bounds": [ [ "-10", "10" ], [ "-10", "10" ] ] }, "candidate": null, "paper_system_number": 9, "paper_system_label": "System 9", "local_name": "S009__DCB-LIT-P1-009" }, { "record_id": "DCB-LIT-P1-010", "problem": { "problem_id": "dcb_lit_p1_010", "task_type": "stability", "name": "DynaCertBench System 010 · DCB-LIT-P1-010 · Table~4 \\#2 (SineModel), \\cite{edwards2024fossil}", "dynamics": "dx1/dt = x2+u1, dx2/dt = u2+((3/20)*(981/100)*(1/2)*sin(x1)-(1/10)*x2)/((3/20)*(1/2)**2)", "source": "paper System 10; canonical record DCB-LIT-P1-010", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-010.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner radius is provenance only; formal N includes the equilibrium." ], "controller_parameters": "u1,u2", "equilibrium": [ 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "constraints", "constraints": [ { "lhs": "x1**2 + x2**2", "relation": "le", "rhs": "(1)**2" } ], "bounds": [ [ "-1", "1" ], [ "-1", "1" ] ] }, "analysis_bounds": [ [ "-1", "1" ], [ "-1", "1" ] ] }, "candidate": null, "paper_system_number": 10, "paper_system_label": "System 10", "local_name": "S010__DCB-LIT-P1-010" }, { "record_id": "DCB-LIT-P1-011", "problem": { "problem_id": "dcb_lit_p1_011", "task_type": "stability", "name": "DynaCertBench System 011 · DCB-LIT-P1-011 · Table~4 \\#14 (DTAhmadi2), \\cite{edwards2024fossil}", "dynamics": "x1[k+1] = 0.687*x1+0.558*x2-0.0001*x1*x2, x2[k+1] = -0.292*x1+0.773*x2", "source": "paper System 11; canonical record DCB-LIT-P1-011", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-011.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner radius is provenance only; formal N includes the equilibrium." ], "equilibrium": [ 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "constraints", "constraints": [ { "lhs": "x1**2 + x2**2", "relation": "le", "rhs": "(1)**2" } ], "bounds": [ [ "-1", "1" ], [ "-1", "1" ] ] }, "analysis_bounds": [ [ "-1", "1" ], [ "-1", "1" ] ] }, "candidate": null, "paper_system_number": 11, "paper_system_label": "System 11", "local_name": "S011__DCB-LIT-P1-011" }, { "record_id": "DCB-LIT-P1-012", "problem": { "problem_id": "dcb_lit_p1_012", "task_type": "stability", "name": "DynaCertBench System 012 · DCB-LIT-P1-012 · Table~4 \\#15 (DTAhmadi), \\cite{edwards2024fossil}", "dynamics": "x1[k+1] = 0.3*x1+u1, x2[k+1] = -x1+0.5*x2+((7)/(18))*x1**2+u2", "source": "paper System 12; canonical record DCB-LIT-P1-012", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-012.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner radius is provenance only; formal N includes the equilibrium." ], "controller_parameters": "u1,u2", "equilibrium": [ 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "constraints", "constraints": [ { "lhs": "x1**2 + x2**2", "relation": "le", "rhs": "(1)**2" } ], "bounds": [ [ "-1", "1" ], [ "-1", "1" ] ] }, "analysis_bounds": [ [ "-1", "1" ], [ "-1", "1" ] ] }, "candidate": null, "paper_system_number": 12, "paper_system_label": "System 12", "local_name": "S012__DCB-LIT-P1-012" }, { "record_id": "DCB-LIT-P1-013", "problem": { "problem_id": "dcb_lit_p1_013", "task_type": "stability", "name": "DynaCertBench System 013 · DCB-LIT-P1-013 · Table~1 row~4, \\cite{zou2025analytical}", "dynamics": "dx1/dt = -3*x1**3+3*x1*x3-9*x1, dx2/dt = -x1**3-5*x2+5*x3**3, dx3/dt = -9*x3**3", "source": "paper System 13; canonical record DCB-LIT-P1-013", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-013.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner hole is retained only as provenance; N is the origin-containing outer box." ], "equilibrium": [ 0, 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "box", "bounds": [ [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ] ] }, "analysis_bounds": [ [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ] ] }, "candidate": null, "paper_system_number": 13, "paper_system_label": "System 13", "local_name": "S013__DCB-LIT-P1-013" }, { "record_id": "DCB-LIT-P1-014", "problem": { "problem_id": "dcb_lit_p1_014", "task_type": "stability", "name": "DynaCertBench System 014 · DCB-LIT-P1-014 · Table~1 row~5, \\cite{zou2025analytical}", "dynamics": "dx1/dt = x2, dx2/dt = -sin(x1)*cos(x1)-x2-sin(x3)*cos(x3), dx3/dt = x2-x3", "source": "paper System 14; canonical record DCB-LIT-P1-014", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-014.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner hole is retained only as provenance; N is the origin-containing outer box." ], "equilibrium": [ 0, 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "box", "bounds": [ [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ] ] }, "analysis_bounds": [ [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ] ] }, "candidate": null, "paper_system_number": 14, "paper_system_label": "System 14", "local_name": "S014__DCB-LIT-P1-014" }, { "record_id": "DCB-LIT-P1-015", "problem": { "problem_id": "dcb_lit_p1_015", "task_type": "stability", "name": "DynaCertBench System 015 · DCB-LIT-P1-015 · Table~1 row~6, \\cite{zou2025analytical}", "dynamics": "dq/dt = w1-w2, dw1/dt = (1/2)*(-1-2*w1-sin(q)+cos(q)), dw2/dt = (1/2)*(-1-2*w2+sin(q)+cos(q))", "source": "paper System 15; canonical record DCB-LIT-P1-015", "assumptions": [ "Audited center-of-inertia reduction with delta1=q/2 and delta2=-q/2.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner hole is retained only as provenance; N is the origin-containing outer box." ], "equilibrium": [ 0, 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "box", "bounds": [ [ "-1.5", "1.5" ], [ "-2", "2" ], [ "-2", "2" ] ] }, "analysis_bounds": [ [ "-1.5", "1.5" ], [ "-2", "2" ], [ "-2", "2" ] ] }, "candidate": null, "paper_system_number": 15, "paper_system_label": "System 15", "local_name": "S015__DCB-LIT-P1-015" }, { "record_id": "DCB-LIT-P1-016", "problem": { "problem_id": "dcb_lit_p1_016", "task_type": "stability", "name": "DynaCertBench System 016 · DCB-LIT-P1-016 · Table~1 row~7, \\cite{zou2025analytical}", "dynamics": "dx1/dt = -x1+0.5*x2-0.1*x5**2, dx2/dt = -0.5*x1-x2, dx3/dt = -x3+0.5*x4-0.1*x1**2, dx4/dt = -0.5*x3-x4, dx5/dt = -x5+0.5*x6, dx6/dt = -0.5*x5-x6+0.1*x2**2", "source": "paper System 16; canonical record DCB-LIT-P1-016", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-016.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner hole is retained only as provenance; N is the origin-containing outer box." ], "equilibrium": [ 0, 0, 0, 0, 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "box", "bounds": [ [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ] ] }, "analysis_bounds": [ [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ] ] }, "candidate": null, "paper_system_number": 16, "paper_system_label": "System 16", "local_name": "S016__DCB-LIT-P1-016" }, { "record_id": "DCB-LIT-P1-017", "problem": { "problem_id": "dcb_lit_p1_017", "task_type": "stability", "name": "DynaCertBench System 017 · DCB-LIT-P1-017 · Table~1 row~8, \\cite{zou2025analytical}", "dynamics": "dx1/dt = x2, dx2/dt = -((3)/(2))*x6*x4-2*x1-((5)/(2))*x2-((1)/(2))*x4*sin(x2)*cos(x4), dx3/dt = x4, dx4/dt = ((3)/(2))*x6*x2-2*x3-10*x4+((1)/(2))*x2*sin(x2)*cos(x4), dx5/dt = x6, dx6/dt = -5*x5-((4)/(5))*x6", "source": "paper System 17; canonical record DCB-LIT-P1-017", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-017.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner hole is retained only as provenance; N is the origin-containing outer box." ], "equilibrium": [ 0, 0, 0, 0, 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "box", "bounds": [ [ "-3", "3" ], [ "-3", "3" ], [ "-3", "3" ], [ "-3", "3" ], [ "-3", "3" ], [ "-3", "3" ] ] }, "analysis_bounds": [ [ "-3", "3" ], [ "-3", "3" ], [ "-3", "3" ], [ "-3", "3" ], [ "-3", "3" ], [ "-3", "3" ] ] }, "candidate": null, "paper_system_number": 17, "paper_system_label": "System 17", "local_name": "S017__DCB-LIT-P1-017" }, { "record_id": "DCB-LIT-P1-018", "problem": { "problem_id": "dcb_lit_p1_018", "task_type": "stability", "name": "DynaCertBench System 018 · DCB-LIT-P1-018 · Table~1 row~9, \\cite{zou2025analytical}", "dynamics": "dd1/dt = w1-(w1+w2+w3)/3, dd2/dt = w2-(w1+w2+w3)/3, dw1/dt = -w1-(1/2)*(sin(d1-d2)+sin(2*d1+d2)), dw2/dt = -w2-(1/2)*(sin(d2-d1)+sin(d1+2*d2)), dw3/dt = -w3+(1/2)*(sin(2*d1+d2)+sin(d1+2*d2))", "source": "paper System 18; canonical record DCB-LIT-P1-018", "assumptions": [ "Audited center-of-inertia reduction with delta3=-(delta1+delta2).", "The retained common-frequency mode decays; angle derivatives use omega_i minus the mean frequency.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner hole is retained only as provenance; N is the origin-containing outer constrained region." ], "equilibrium": [ 0, 0, 0, 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "constraints", "bounds": [ [ "-0.75", "0.75" ], [ "-0.75", "0.75" ], [ "-1.2", "1.2" ], [ "-1.2", "1.2" ], [ "-1.2", "1.2" ] ], "constraints": [ { "lhs": "d1+d2", "relation": "le", "rhs": "0.75" }, { "lhs": "d1+d2", "relation": "ge", "rhs": "-0.75" } ] }, "analysis_bounds": [ [ "-0.75", "0.75" ], [ "-0.75", "0.75" ], [ "-1.2", "1.2" ], [ "-1.2", "1.2" ], [ "-1.2", "1.2" ] ] }, "candidate": null, "paper_system_number": 18, "paper_system_label": "System 18", "local_name": "S018__DCB-LIT-P1-018" }, { "record_id": "DCB-LIT-P1-019", "problem": { "problem_id": "dcb_lit_p1_019", "task_type": "stability", "name": "DynaCertBench System 019 · DCB-LIT-P1-019 · Table~1 row~10, \\cite{zou2025analytical}", "dynamics": "dx1/dt = -x1+0.5*x2-0.1*x5**2, dx2/dt = -0.5*x1-x2+0.1*x8, dx3/dt = -x3+0.5*x4-0.1*x1**2, dx4/dt = -0.5*x3-x4, dx5/dt = -x5+0.5*x6, dx6/dt = -0.5*x5-x6+0.1*x2**2, dx7/dt = x8, dx8/dt = -sin(x7)*cos(x7)-x8-sin(x9)*cos(x9)-0.1*x2, dx9/dt = x8-x9", "source": "paper System 19; canonical record DCB-LIT-P1-019", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-019.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner hole is retained only as provenance; N is the origin-containing outer box." ], "equilibrium": [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "box", "bounds": [ [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ] ] }, "analysis_bounds": [ [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ], [ "-1.5", "1.5" ] ] }, "candidate": null, "paper_system_number": 19, "paper_system_label": "System 19", "local_name": "S019__DCB-LIT-P1-019" }, { "record_id": "DCB-LIT-P1-020", "problem": { "problem_id": "dcb_lit_p1_020", "task_type": "stability", "name": "DynaCertBench System 020 · DCB-LIT-P1-020 · Appendix Table~6, \\cite{zou2025analytical}", "dynamics": "dx1/dt = -8*x1*x2**2-10*x2**4, dx2/dt = -8*x2**3+3*x2**2-8*x2, dx3/dt = -x3", "source": "paper System 20; canonical record DCB-LIT-P1-020", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-020.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source epsilon inner hole and epsilon-stability label are retained only as provenance; the formal task uses the origin-containing outer box and ordinary local Lyapunov stability." ], "equilibrium": [ 0, 0, 0 ], "stability_kind": "local_lyapunov", "neighborhood": { "type": "box", "bounds": [ [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ] ] }, "analysis_bounds": [ [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ] ] }, "candidate": null, "paper_system_number": 20, "paper_system_label": "System 20", "local_name": "S020__DCB-LIT-P1-020" }, { "record_id": "DCB-LIT-P1-021", "problem": { "problem_id": "dcb_lit_p1_021", "task_type": "stability", "name": "DynaCertBench System 021 · DCB-LIT-P1-021 · Main theorem, \\cite{liu2026globally}", "dynamics": "dx1/dt = 4*x1**3-x1**2*x2-6*x1*x2**2-x2**3, dx2/dt = x1**3+4*x1**2*x2+x1*x2**2-6*x2**3", "source": "paper System 21; canonical record DCB-LIT-P1-021", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P1-021.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "equilibrium": [ 0, 0 ], "stability_kind": "global_asymptotic", "neighborhood": { "type": "unbounded" }, "analysis_bounds": [ [ -10, 10 ], [ -10, 10 ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 21, "paper_system_label": "System 21", "local_name": "S021__DCB-LIT-P1-021" }, { "record_id": "DCB-LIT-P2-001", "problem": { "problem_id": "dcb_lit_p2_001", "task_type": "roa", "name": "DynaCertBench System 022 · DCB-LIT-P2-001 · Section~5.3 and Appendix~C.3.3, \\cite{zhou2022neural}", "dynamics": "dx1/dt = x2, dx2/dt = ((3/20)*(981/100)*(1/2)*sin(x1)+(20*tanh(-(23286329/1000000)*x1-(105411/20000)*x2))-(1/10)*x2)/((3/20)*(1/2)**2)", "source": "paper System 22; canonical record DCB-LIT-P2-001", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P2-001.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "equilibrium": [ 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "ball", "center": [ "0", "0" ], "radius": "4" }, "analysis_bounds": [ [ "-4", "4" ], [ "-4", "4" ] ] }, "candidate": null, "paper_system_number": 22, "paper_system_label": "System 22", "local_name": "S022__DCB-LIT-P2-001" }, { "record_id": "DCB-LIT-P2-002", "problem": { "problem_id": "dcb_lit_p2_002", "task_type": "roa", "name": "DynaCertBench System 023 · DCB-LIT-P2-002 · Section~5.1 and Appendix~C.3.1, \\cite{zhou2022neural}", "dynamics": "dx1/dt = -x2, dx2/dt = x1 + (x1**2 - 1)*x2", "equilibrium": [ 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "ball", "center": [ 0, 0 ], "radius": "6/5" }, "analysis_bounds": [ [ "-6/5", "6/5" ], [ "-6/5", "6/5" ] ], "source": "paper System 23; canonical record DCB-LIT-P2-002", "assumptions": [ "The formal neighborhood is the closed ball x1^2+x2^2 <= (6/5)^2.", "No separate initial set is supplied; the target is to certify Omega_c itself." ] }, "candidate": null, "paper_system_number": 23, "paper_system_label": "System 23", "local_name": "S023__DCB-LIT-P2-002" }, { "record_id": "DCB-LIT-P2-003", "problem": { "problem_id": "dcb_lit_p2_003", "task_type": "roa", "name": "DynaCertBench System 024 · DCB-LIT-P2-003 · Example~3.1, \\cite{liu2023towards}", "dynamics": "dx/dt = -x + x**3", "equilibrium": [ 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "constraint", "lhs": "x**2", "relation": "lt", "rhs": "1", "bounds": [ [ -1, 1 ] ] }, "analysis_bounds": [ [ -1, 1 ] ], "source": "paper System 24; canonical record DCB-LIT-P2-003", "assumptions": [ "The strict constraint x^2 < 1 represents the source neighborhood (-1,1).", "No separate initial set is supplied; the target is to certify Omega_c itself." ] }, "candidate": null, "paper_system_number": 24, "paper_system_label": "System 24", "local_name": "S024__DCB-LIT-P2-003" }, { "record_id": "DCB-LIT-P2-004", "problem": { "problem_id": "dcb_lit_p2_004", "task_type": "roa", "name": "DynaCertBench System 025 · DCB-LIT-P2-004 · Example~6.2 and Table~II, \\cite{liu2023towards}", "dynamics": "dx1/dt = x2, dx2/dt = -2*x1 + 1/3*x1**3 - x2", "equilibrium": [ 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "box", "bounds": [ [ -6, 6 ], [ -6, 6 ] ] }, "analysis_bounds": [ [ -6, 6 ], [ -6, 6 ] ], "source": "paper System 25; canonical record DCB-LIT-P2-004", "assumptions": [ "No separate initial set is supplied; the target is to certify Omega_c itself." ] }, "candidate": null, "paper_system_number": 25, "paper_system_label": "System 25", "local_name": "S025__DCB-LIT-P2-004" }, { "record_id": "DCB-LIT-P2-005", "problem": { "problem_id": "dcb_lit_p2_005", "task_type": "roa", "name": "DynaCertBench System 026 · DCB-LIT-P2-005 · Table~4 \\#3 (LorenzSystem), \\cite{edwards2024fossil}", "dynamics": "dx1/dt = -10*(x1-x2) + u1, dx2/dt = 28*x1 - x2 - x1*x3 + u2, dx3/dt = x1*x2 - 8/3*x3 + u3", "controller_parameters": "u1,u2,u3", "equilibrium": [ 0, 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "ball", "center": [ 0, 0, 0 ], "radius": 2 }, "initial_set": { "type": "ball", "center": [ 0, 0, 0 ], "radius": "3/10" }, "analysis_bounds": [ [ -2, 2 ], [ -2, 2 ], [ -2, 2 ] ], "source": "paper System 26; canonical record DCB-LIT-P2-005", "assumptions": [ "The source leaves u1,u2,u3 unconstrained over the reals, so no artificial control box is imposed.", "The supplied initial ball must be contained in the synthesized Omega_c." ] }, "candidate": null, "paper_system_number": 26, "paper_system_label": "System 26", "local_name": "S026__DCB-LIT-P2-005" }, { "record_id": "DCB-LIT-P2-006", "paper_system_number": 27, "paper_system_label": "System 27", "local_name": "S027__DCB-LIT-P2-006", "problem": { "problem_id": "dcb_lit_p2_006", "task_type": "roa", "name": "DynaCertBench System 027 · DCB-LIT-P2-006 · Example~5.2 and Table~2, \\cite{liu2024tool}", "dynamics": "dx1/dt = x2, dx2/dt = -1/2*x2-sin(x1+pi/3)+sin(pi/3)", "source": "paper System 27; canonical record DCB-LIT-P2-006", "assumptions": [ "Transcribed from paper-4-20-20260904 System 27.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The final sine offset is positive in PDF 4-20, making the origin an equilibrium." ], "analysis_bounds": [ [ "-2", "3" ], [ "-3", "3/2" ] ], "equilibrium": [ 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "box", "bounds": [ [ "-2", "3" ], [ "-3", "3/2" ] ] } }, "candidate": null }, { "record_id": "DCB-LIT-P2-007", "problem": { "problem_id": "dcb_lit_p2_007", "task_type": "roa", "name": "DynaCertBench System 028 · DCB-LIT-P2-007 · Example~5.4 and Table~3, \\cite{liu2024tool}", "dynamics": "dx1/dt = -x1 + 1/2*x2 - 1/10*x9**2, dx2/dt = -1/2*x1 - x2, dx3/dt = -x3 + 1/2*x4 - 1/10*x1**2, dx4/dt = -1/2*x3 - x4, dx5/dt = -x5 + 1/2*x6 + 1/10*x7**2, dx6/dt = -1/2*x5 - x6, dx7/dt = -x7 + 1/2*x8, dx8/dt = -1/2*x7 - x8, dx9/dt = -x9 + 1/2*x10, dx10/dt = -1/2*x9 - x10 + 1/10*x2**2", "equilibrium": [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "box", "bounds": [ [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ] ] }, "analysis_bounds": [ [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ] ], "source": "paper System 28; canonical record DCB-LIT-P2-007", "assumptions": [ "No separate initial set is supplied; the target is to certify Omega_c itself." ] }, "candidate": null, "paper_system_number": 28, "paper_system_label": "System 28", "local_name": "S028__DCB-LIT-P2-007" }, { "record_id": "DCB-LIT-P2-008", "problem": { "problem_id": "dcb_lit_p2_008", "task_type": "roa", "name": "DynaCertBench System 029 · DCB-LIT-P2-008 · Section~5.1, with verification details in Appendix~E.2 (\\(n=1\\)), \\cite{meng2024physics}", "dynamics": "dx/dt = -x - x**3", "equilibrium": [ 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "box", "bounds": [ [ -1, 1 ] ] }, "analysis_bounds": [ [ -1, 1 ] ], "source": "paper System 29; canonical record DCB-LIT-P2-008", "assumptions": [ "No separate initial set is supplied; the target is to certify Omega_c itself." ] }, "candidate": null, "paper_system_number": 29, "paper_system_label": "System 29", "local_name": "S029__DCB-LIT-P2-008" }, { "record_id": "DCB-LIT-P3-001", "problem": { "problem_id": "dcb_lit_p3_001", "task_type": "invariant", "name": "DynaCertBench System 030 · DCB-LIT-P3-001 · Appendix~A.1 (PVTOL), \\cite{wu2023neural}", "dynamics": "x1[k+1] = x1+1/20*(x4*cos(x3)-x5*sin(x3)), x2[k+1] = x2+1/20*(x4*sin(x3)+x5*cos(x3)), x3[k+1] = x3+1/20*x6, x4[k+1] = x4+1/20*(x5*x6-49/5*sin(x3)), x5[k+1] = x5+1/20*(-x4*x6-49/5*cos(x3)+(u1+u2)/4), x6[k+1] = x6+1/20*((1/4)/(19/400)*(u1-u2))", "source": "paper System 30; canonical record DCB-LIT-P3-001", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P3-001.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u1,u2", "control_bounds": [ [ 0, "196/5" ], [ 0, "196/5" ] ], "unsafe_set": { "type": "box", "bounds": [ [ "-(1)", "1" ], [ "-(1)", "1" ], [ "-(1)", "1" ], [ "-(1)", "1" ], [ "-(1)", "1" ], [ "-(1)", "1" ] ], "complement": true }, "analysis_bounds": [ [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ] ], "safety_interpretation": "outside_X_is_unsafe" }, "candidate": null, "paper_system_number": 30, "paper_system_label": "System 30", "local_name": "S030__DCB-LIT-P3-001" }, { "record_id": "DCB-LIT-P3-002", "problem": { "problem_id": "dcb_lit_p3_002", "task_type": "invariant", "name": "DynaCertBench System 031 · DCB-LIT-P3-002 · Example~1, \\cite{samanipour2026relu}", "dynamics": "dx1/dt = x2, dx2/dt = sin(x1)-3*x1-3*x2", "source": "paper System 31; canonical record DCB-LIT-P3-002", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P3-002.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The paper fixes u=-3*x1-3*x2; it is substituted into the closed-loop dynamics, not synthesized as a free input." ], "unsafe_set": { "type": "box", "bounds": [ [ "-(pi)", "pi" ], [ "-(pi)", "pi" ] ], "complement": true }, "analysis_bounds": [ [ "-4", "4" ], [ "-4", "4" ] ], "safety_interpretation": "outside_X_is_unsafe" }, "candidate": null, "paper_system_number": 31, "paper_system_label": "System 31", "local_name": "S031__DCB-LIT-P3-002" }, { "record_id": "DCB-LIT-P3-003", "problem": { "problem_id": "dcb_lit_p3_003", "task_type": "invariant", "name": "DynaCertBench System 032 · DCB-LIT-P3-003 · Example~2, \\cite{samanipour2026relu}", "dynamics": "dx1/dt = x1**2+x1*x2+x1, dx2/dt = x1*x2+x2**2+x2", "source": "paper System 32; canonical record DCB-LIT-P3-003", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P3-003.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "unsafe_set": { "type": "box", "bounds": [ [ "-1", "0" ], [ "-1", "0" ] ] }, "analysis_bounds": [ [ "-2", "2" ], [ "-2", "2" ] ], "safety_interpretation": "conditional_in_X" }, "candidate": null, "paper_system_number": 32, "paper_system_label": "System 32", "local_name": "S032__DCB-LIT-P3-003" }, { "record_id": "DCB-LIT-P3-004", "problem": { "problem_id": "dcb_lit_p3_004", "task_type": "invariant", "name": "DynaCertBench System 033 · DCB-LIT-P3-004 · Example~3, \\cite{samanipour2026relu}", "dynamics": "dx1/dt = x2, dx2/dt = ((x1+2.4109*x2+34.3620*x3+10.7009*x4)+(1/10)*x4**2*sin(x3)-(981/1000)*sin(x3)*cos(x3))/(1+(1/10)*sin(x3)**2), dx3/dt = x4, dx4/dt = ((10791/1000)*sin(x3)-(x1+2.4109*x2+34.3620*x3+10.7009*x4)*cos(x3)-(1/10)*x4**2*sin(x3)*cos(x3))/(1+(1/10)*sin(x3)**2)", "source": "paper System 33; canonical record DCB-LIT-P3-004", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P3-004.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The paper's fixed linear policy is substituted exactly into the closed-loop dynamics; it is not synthesized as a free input." ], "unsafe_set": { "type": "box", "bounds": [ [ "-(1)", "1" ], [ "-(1)", "1" ], [ "-(1)", "1" ], [ "-(1)", "1" ] ], "complement": true }, "analysis_bounds": [ [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ], [ "-1", "1" ] ], "safety_interpretation": "outside_X_is_unsafe" }, "candidate": null, "paper_system_number": 33, "paper_system_label": "System 33", "local_name": "S033__DCB-LIT-P3-004" }, { "record_id": "DCB-LIT-P3-005", "problem": { "problem_id": "dcb_lit_p3_005", "task_type": "invariant", "name": "DynaCertBench System 034 · DCB-LIT-P3-005 · Section~6.1.1, \\cite{vaisi2025extracting}", "dynamics": "x1[k+1] = x1+0.01*x2, x2[k+1] = x2+0.01*(9.8*sin(x1)+u)", "source": "paper System 34; canonical record DCB-LIT-P3-005", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P3-005.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u", "unsafe_set": { "type": "box", "bounds": [ [ "-(pi/4)", "pi/4" ], [ "-(pi/4)", "pi/4" ] ], "complement": true }, "analysis_bounds": [ [ "-1", "1" ], [ "-1", "1" ] ], "safety_interpretation": "outside_X_is_unsafe", "control_bounds": [ [ -10, 10 ] ] }, "candidate": null, "paper_system_number": 34, "paper_system_label": "System 34", "local_name": "S034__DCB-LIT-P3-005" }, { "record_id": "DCB-LIT-P3-006", "problem": { "problem_id": "dcb_lit_p3_006", "task_type": "invariant", "name": "DynaCertBench System 035 · DCB-LIT-P3-006 · Section~6.1.2, \\cite{vaisi2025extracting}", "dynamics": "x1[k+1] = x2, x2[k+1] = 29.4*sin(x1)+300*cos(x1)*tan(x3)+15*u*cos(x1)*cos(x3)**2, x3[k+1] = u", "source": "paper System 35; canonical record DCB-LIT-P3-006", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P3-006.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u", "unsafe_set": { "type": "box", "bounds": [ [ "-(11/5)", "11/5" ], [ "-(11/5)", "11/5" ], [ "-(11/5)", "11/5" ] ], "complement": true }, "analysis_bounds": [ [ "-11/5", "11/5" ], [ "-11/5", "11/5" ], [ "-11/5", "11/5" ] ], "safety_interpretation": "outside_X_is_unsafe", "control_bounds": [ [ -10, 10 ] ] }, "candidate": null, "paper_system_number": 35, "paper_system_label": "System 35", "local_name": "S035__DCB-LIT-P3-006" }, { "record_id": "DCB-LIT-P4-001", "problem": { "problem_id": "dcb_lit_p4_001", "task_type": "safety", "name": "DynaCertBench System 036 · DCB-LIT-P4-001 · Theorem~4, \\cite{sogokon2018vector}", "dynamics": "dx1/dt = x2, dx2/dt = x1", "source": "paper System 36; canonical record DCB-LIT-P4-001", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-001.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "constraints", "constraints": [ { "lhs": "x1", "relation": "le", "rhs": "0" }, { "lhs": "x2", "relation": "le", "rhs": "0" } ] }, "unsafe_set": { "type": "constraint", "lhs": "x1", "relation": "gt", "rhs": "0" }, "analysis_bounds": [ [ -3, 3 ], [ -3, 3 ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 36, "paper_system_label": "System 36", "local_name": "S036__DCB-LIT-P4-001" }, { "record_id": "DCB-LIT-P4-002", "problem": { "problem_id": "dcb_lit_p4_002", "task_type": "safety", "name": "DynaCertBench System 037 · DCB-LIT-P4-002 · Example~1, \\cite{sogokon2018vector}", "dynamics": "dx1/dt = 2*x1+x2+2*x3-x1**2+x1*x3-x3**2, dx2/dt = -2+x1-x2-x2**2, dx3/dt = -2-x2-x3+x1**2-x1*x3", "source": "paper System 37; canonical record DCB-LIT-P4-002", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-002.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "constraints", "constraints": [ { "lhs": "x1", "relation": "le", "rhs": "0" }, { "lhs": "x2", "relation": "le", "rhs": "0" }, { "lhs": "x3", "relation": "le", "rhs": "0" } ] }, "unsafe_set": { "type": "constraint", "lhs": "x1+x2+x3", "relation": "ge", "rhs": "1" }, "analysis_bounds": [ [ -3, 3 ], [ -3, 3 ], [ -3, 3 ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 37, "paper_system_label": "System 37", "local_name": "S037__DCB-LIT-P4-002" }, { "record_id": "DCB-LIT-P4-003", "problem": { "problem_id": "dcb_lit_p4_003", "task_type": "safety", "name": "DynaCertBench System 038 · DCB-LIT-P4-003 · Example~3.1, \\cite{zhao2020synthesizing}", "dynamics": "dx1/dt = x2, dx2/dt = -x1+((1)/(3))*x1**3-x2", "source": "paper System 38; canonical record DCB-LIT-P4-003", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-003.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "ball", "center": [ "3/2", "0" ], "radius": "1/2" }, "unsafe_set": { "type": "ball", "center": [ "-1", "-1" ], "radius": "2/5" }, "analysis_bounds": [ [ "-7/2", "2" ], [ "-2", "1" ] ] }, "candidate": null, "paper_system_number": 38, "paper_system_label": "System 38", "local_name": "S038__DCB-LIT-P4-003" }, { "record_id": "DCB-LIT-P4-004", "problem": { "problem_id": "dcb_lit_p4_004", "task_type": "safety", "name": "DynaCertBench System 039 · DCB-LIT-P4-004 · Table~1 row~9 (Barr1), \\cite{abate2021fossil}", "dynamics": "dx1/dt = x2+2*x1*x2, dx2/dt = -x1-x2**2+2*x1**2", "source": "paper System 39; canonical record DCB-LIT-P4-004", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-004.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "box", "bounds": [ [ "0", "1" ], [ "1", "2" ] ] }, "unsafe_set": { "type": "constraint", "lhs": "x1+x2**2", "relation": "le", "rhs": "0", "bounds": [ [ "-2", "2" ], [ "-2", "2" ] ] }, "analysis_bounds": [ [ "-2", "2" ], [ "-2", "2" ] ] }, "candidate": null, "paper_system_number": 39, "paper_system_label": "System 39", "local_name": "S039__DCB-LIT-P4-004" }, { "record_id": "DCB-LIT-P4-005", "problem": { "problem_id": "dcb_lit_p4_005", "task_type": "safety", "name": "DynaCertBench System 040 · DCB-LIT-P4-005 · Table~1 row~12 (Barr4), \\cite{abate2021fossil}", "dynamics": "dx1/dt = sin(x3), dx2/dt = cos(x3), dx3/dt = -sin(x3)+((3*(x1*sin(x3)+x2*cos(x3)))/(0.5+x1**2+x2**2))", "source": "paper System 40; canonical record DCB-LIT-P4-005", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-005.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "box", "bounds": [ [ "-1/10", "1/10" ], [ "-2", "-9/5" ], [ "-13/25", "13/25" ] ] }, "unsafe_set": { "type": "constraint", "lhs": "x1**2+x2**2", "relation": "le", "rhs": "1/25", "bounds": [ [ "-2", "2" ], [ "-2", "2" ], [ "-157/100", "157/100" ] ] }, "analysis_bounds": [ [ "-2", "2" ], [ "-2", "2" ], [ "-157/100", "157/100" ] ] }, "candidate": null, "paper_system_number": 40, "paper_system_label": "System 40", "local_name": "S040__DCB-LIT-P4-005" }, { "record_id": "DCB-LIT-P4-006", "problem": { "problem_id": "dcb_lit_p4_006", "task_type": "safety", "name": "DynaCertBench System 041 · DCB-LIT-P4-006 · Table~1 row~17 (Hi-Ord8), \\cite{abate2021fossil}", "dynamics": "dx0/dt = x1, dx1/dt = x2, dx2/dt = x3, dx3/dt = x4, dx4/dt = x5, dx5/dt = x6, dx6/dt = x7, dx7/dt = -(20*x7 + 170*x6 + 800*x5 + 2273*x4 + 3980*x3 + 4180*x2 + 2400*x1 + 576*x0)", "initial_set": { "type": "box", "bounds": [ [ "9/10", "11/10" ], [ "9/10", "11/10" ], [ "9/10", "11/10" ], [ "9/10", "11/10" ], [ "9/10", "11/10" ], [ "9/10", "11/10" ], [ "9/10", "11/10" ], [ "9/10", "11/10" ] ] }, "unsafe_set": { "type": "box", "bounds": [ [ "-11/5", "-9/5" ], [ "-11/5", "-9/5" ], [ "-11/5", "-9/5" ], [ "-11/5", "-9/5" ], [ "-11/5", "-9/5" ], [ "-11/5", "-9/5" ], [ "-11/5", "-9/5" ], [ "-11/5", "-9/5" ] ] }, "analysis_bounds": [ [ "-11/5", "11/5" ], [ "-11/5", "11/5" ], [ "-11/5", "11/5" ], [ "-11/5", "11/5" ], [ "-11/5", "11/5" ], [ "-11/5", "11/5" ], [ "-11/5", "11/5" ], [ "-11/5", "11/5" ] ], "source": "paper System 41; canonical record DCB-LIT-P4-006", "assumptions": [ "The working domain is X=[-11/5, 11/5]^8.", "The initial set is [9/10, 11/10]^8.", "The unsafe set is [-11/5, -9/5]^8." ] }, "candidate": null, "paper_system_number": 41, "paper_system_label": "System 41", "local_name": "S041__DCB-LIT-P4-006" }, { "record_id": "DCB-LIT-P4-007", "problem": { "problem_id": "dcb_lit_p4_007", "task_type": "safety", "name": "DynaCertBench System 042 · DCB-LIT-P4-007 · Table~1 (arch2[36], instance~1), \\cite{wu2024completeness}", "dynamics": "dx1/dt = x1**2+x2**2-1, dx2/dt = 5*(x1*x2-1)", "source": "paper System 42; canonical record DCB-LIT-P4-007", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-007.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "ball", "center": [ "0", "2" ], "radius": "3/2" }, "unsafe_set": { "type": "box", "bounds": [ [ "-3", "-1" ], [ "-3", "-2" ] ] }, "analysis_bounds": [ [ -4, 4 ], [ -4, 4 ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 42, "paper_system_label": "System 42", "local_name": "S042__DCB-LIT-P4-007" }, { "record_id": "DCB-LIT-P4-008", "problem": { "problem_id": "dcb_lit_p4_008", "task_type": "safety", "name": "DynaCertBench System 043 · DCB-LIT-P4-008 · Table~1 (arch3[36], instance~1), \\cite{wu2024completeness}", "dynamics": "dx1/dt = x1-x1**3+x2-x1*x2**2, dx2/dt = -x1+x2-x1**2*x2-x2**3", "source": "paper System 43; canonical record DCB-LIT-P4-008", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-008.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "ball", "center": [ "0", "0" ], "radius": "1/5" }, "unsafe_set": { "type": "ball", "center": [ "5/2", "5/2" ], "radius": "1/2" }, "analysis_bounds": [ [ -1, 4 ], [ -1, 4 ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 43, "paper_system_label": "System 43", "local_name": "S043__DCB-LIT-P4-008" }, { "record_id": "DCB-LIT-P4-009", "problem": { "problem_id": "dcb_lit_p4_009", "task_type": "safety", "name": "DynaCertBench System 044 · DCB-LIT-P4-009 · Example~5 and Table~1, \\cite{liu2015abstraction}", "dynamics": "dx1/dt = exp(-x1)+x2-1, dx2/dt = -sin(x1)**2", "source": "paper System 44; canonical record DCB-LIT-P4-009", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-009.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "ball", "center": [ "-1/2", "1/2" ], "radius": "2/5" }, "unsafe_set": { "type": "ball", "center": [ "7/10", "-7/10" ], "radius": "3/10" }, "analysis_bounds": [ [ -2, 2 ], [ -2, 2 ] ] }, "candidate": null, "paper_system_number": 44, "paper_system_label": "System 44", "local_name": "S044__DCB-LIT-P4-009" }, { "record_id": "DCB-LIT-P4-010", "problem": { "problem_id": "dcb_lit_p4_010", "task_type": "safety", "name": "DynaCertBench System 045 · DCB-LIT-P4-010 · Example~6 and Table~1, \\cite{liu2015abstraction}", "dynamics": "dx1/dt = -(((1/5)*10*x1*x2)/(x1+x2+x3))-(1/125)*x1, dx2/dt = (((1/5)*10*x1*x2)/(x1+x2+x3))-((1/125)+(1/10))*x2, dx3/dt = 1/10*x2-19/20*x3", "source": "paper System 45; canonical record DCB-LIT-P4-010", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-010.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "box", "bounds": [ [ "1997/200", "1999/200" ], [ "1/200", "3/200" ], [ "0", "3/1000" ] ] }, "unsafe_set": { "type": "constraint", "lhs": "x3", "relation": "gt", "rhs": "1" }, "analysis_bounds": [ [ 0, "10013/1000" ], [ 0, "10013/1000" ], [ 0, "10013/1000" ] ] }, "candidate": null, "paper_system_number": 45, "paper_system_label": "System 45", "local_name": "S045__DCB-LIT-P4-010" }, { "record_id": "DCB-LIT-P4-011", "problem": { "problem_id": "dcb_lit_p4_011", "task_type": "safety", "name": "DynaCertBench System 046 · DCB-LIT-P4-011 · Table~1 (lotka[14], instance~1), \\cite{wu2024completeness}", "dynamics": "dx1/dt = x1*(1-x3), dx2/dt = x2*(1-2*x3), dx3/dt = x3*(x1+x2-1)", "source": "paper System 46; canonical record DCB-LIT-P4-011", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-011.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "ball", "center": [ "1", "1", "3/2" ], "radius": "3/10" }, "unsafe_set": { "type": "ball", "center": [ "-1", "-1", "-3/2" ], "radius": "3/10" }, "analysis_bounds": [ [ -3, 3 ], [ -3, 3 ], [ -3, 3 ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 46, "paper_system_label": "System 46", "local_name": "S046__DCB-LIT-P4-011" }, { "record_id": "DCB-LIT-P4-012", "problem": { "problem_id": "dcb_lit_p4_012", "task_type": "safety", "name": "DynaCertBench System 047 · DCB-LIT-P4-012 · Table~4 \\#5 (ctrlbarr obstacle), \\cite{edwards2024fossil}", "dynamics": "dx1/dt = sin(x3), dx2/dt = cos(x3), dx3/dt = -sin(x3)+u1", "source": "paper System 47; canonical record DCB-LIT-P4-012", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-012.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u1", "initial_set": { "type": "box", "bounds": [ [ "4", "6" ], [ "4", "6" ], [ "-pi/2", "pi/2" ] ] }, "unsafe_set": { "type": "box", "bounds": [ [ "-9", "-7" ], [ "-9", "-6" ], [ "-pi/2", "pi/2" ] ] }, "analysis_bounds": [ [ -10, 10 ], [ -10, 10 ], [ -4, 4 ] ] }, "candidate": null, "paper_system_number": 47, "paper_system_label": "System 47", "local_name": "S047__DCB-LIT-P4-012" }, { "record_id": "DCB-LIT-P4-013", "problem": { "problem_id": "dcb_lit_p4_013", "task_type": "safety", "name": "DynaCertBench System 048 · DCB-LIT-P4-013 · Table~4 \\#17 (ctrlbarr 2roomtemp DT), \\cite{edwards2024fossil}", "dynamics": "x1[k+1] = 0.725*x1+0.25*x2+0.375+0.018*(55-x1)*u1, x2[k+1] = 0.25*x1+0.71*x2+0.6+0.018*(55-x2)*u2", "source": "paper System 48; canonical record DCB-LIT-P4-013", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-013.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u1,u2", "initial_set": { "type": "box", "bounds": [ [ "17", "18" ], [ "17", "18" ] ] }, "unsafe_set": { "type": "box", "bounds": [ [ "28", "30" ], [ "28", "30" ] ] }, "analysis_bounds": [ [ 17, 30 ], [ 17, 30 ] ] }, "candidate": null, "paper_system_number": 48, "paper_system_label": "System 48", "local_name": "S048__DCB-LIT-P4-013" }, { "record_id": "DCB-LIT-P4-014", "problem": { "problem_id": "dcb_lit_p4_014", "task_type": "safety", "name": "DynaCertBench System 049 · DCB-LIT-P4-014 · ~\\texttt{S001}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = x2+u0, dx2/dt = -0.8*x1-0.15*x2+0.2*x1**(2)+u1, dx3/dt = -0.1*x3+0.1*x1+u2", "source": "paper System 49; canonical record DCB-LIT-P4-014", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-014.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1,u2", "initial_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "0.4" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "3", "complement": true }, "analysis_bounds": [ [ "-4", "4" ], [ "-4", "4" ], [ "-4", "4" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 49, "paper_system_label": "System 49", "local_name": "S049__DCB-LIT-P4-014" }, { "record_id": "DCB-LIT-P4-015", "problem": { "problem_id": "dcb_lit_p4_015", "task_type": "safety", "name": "DynaCertBench System 050 · DCB-LIT-P4-015 · ~\\texttt{S002}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = x2+u0, dx2/dt = -0.95*sin(x1)-0.02*x2+0.01*x3+u1, dx3/dt = -0.1*x3+0.02*x1+u2, dx4/dt = -0.03*x4+0.01*x2+u3", "source": "paper System 50; canonical record DCB-LIT-P4-015", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-015.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1,u2,u3", "initial_set": { "type": "ball", "center": [ "0", "0", "0", "0" ], "radius": "0.3" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0", "0" ], "radius": "2.5", "complement": true }, "analysis_bounds": [ [ "-7/2", "7/2" ], [ "-7/2", "7/2" ], [ "-7/2", "7/2" ], [ "-7/2", "7/2" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 50, "paper_system_label": "System 50", "local_name": "S050__DCB-LIT-P4-015" }, { "record_id": "DCB-LIT-P4-016", "problem": { "problem_id": "dcb_lit_p4_016", "task_type": "safety", "name": "DynaCertBench System 051 · DCB-LIT-P4-016 · ~\\texttt{S004}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = 10*(x2-x1)+u0, dx2/dt = 28*x1-x2-x1*x3+u1, dx3/dt = x1*x2-((8)/(3))*x3+u2, dx4/dt = -x2*x3+2.7*x4+u3", "source": "paper System 51; canonical record DCB-LIT-P4-016", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-016.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1,u2,u3", "initial_set": { "type": "ball", "center": [ "0", "0", "0", "0" ], "radius": "0.1" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0", "0" ], "radius": "25", "complement": true }, "analysis_bounds": [ [ "-26", "26" ], [ "-26", "26" ], [ "-26", "26" ], [ "-26", "26" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 51, "paper_system_label": "System 51", "local_name": "S051__DCB-LIT-P4-016" }, { "record_id": "DCB-LIT-P4-017", "problem": { "problem_id": "dcb_lit_p4_017", "task_type": "safety", "name": "DynaCertBench System 052 · DCB-LIT-P4-017 · ~\\texttt{S008}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = -2*x1+6.7*x2-x2*x3+u0, dx2/dt = x1+u1, dx3/dt = -x3+x2**(2)+u2", "source": "paper System 52; canonical record DCB-LIT-P4-017", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-017.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1,u2", "initial_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "0.2" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "5", "complement": true }, "analysis_bounds": [ [ "-6", "6" ], [ "-6", "6" ], [ "-6", "6" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 52, "paper_system_label": "System 52", "local_name": "S052__DCB-LIT-P4-017" }, { "record_id": "DCB-LIT-P4-018", "problem": { "problem_id": "dcb_lit_p4_018", "task_type": "safety", "name": "DynaCertBench System 053 · DCB-LIT-P4-018 · ~\\texttt{S011}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = x2+u0, dx2/dt = -0.3*x1-0.2*(1+cos(x1))*x2-0.1*x1**(3)+u1", "source": "paper System 53; canonical record DCB-LIT-P4-018", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-018.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1", "initial_set": { "type": "ball", "center": [ "0", "0" ], "radius": "0.6" }, "unsafe_set": { "type": "ball", "center": [ "2", "2" ], "radius": "1" }, "analysis_bounds": [ [ "-8/5", "4" ], [ "-8/5", "4" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 53, "paper_system_label": "System 53", "local_name": "S053__DCB-LIT-P4-018" }, { "record_id": "DCB-LIT-P4-019", "problem": { "problem_id": "dcb_lit_p4_019", "task_type": "safety", "name": "DynaCertBench System 054 · DCB-LIT-P4-019 · ~\\texttt{S019}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = -x1+x2*x3, dx2/dt = -x2+x1*x3, dx3/dt = -x3-x1*x2", "source": "paper System 54; canonical record DCB-LIT-P4-019", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-019.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "0.2" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "1", "complement": true }, "analysis_bounds": [ [ "-2", "2" ], [ "-2", "2" ], [ "-2", "2" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 54, "paper_system_label": "System 54", "local_name": "S054__DCB-LIT-P4-019" }, { "record_id": "DCB-LIT-P4-020", "problem": { "problem_id": "dcb_lit_p4_020", "task_type": "safety", "name": "DynaCertBench System 055 · DCB-LIT-P4-020 · ~\\texttt{S024}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = x2+u0, dx2/dt = -x1-0.4*x2+0.15*sin(x3)+u1, dx3/dt = x4+u2, dx4/dt = -0.6*x3-0.3*x4+u3", "source": "paper System 55; canonical record DCB-LIT-P4-020", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-020.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1,u2,u3", "initial_set": { "type": "box", "bounds": [ [ "0.5", "0.8" ], [ "0.3", "0.6" ], [ "0.5", "0.8" ], [ "0.3", "0.6" ] ] }, "unsafe_set": { "type": "union", "sets": [ { "type": "box", "bounds": [ [ "2.5", "3.5" ], [ "2", "3" ], [ "2.5", "3.5" ], [ "2", "3" ] ] }, { "type": "box", "bounds": [ [ "-3.5", "-2.5" ], [ "-3", "-2" ], [ "-3.5", "-2.5" ], [ "-3", "-2" ] ] } ] }, "analysis_bounds": [ [ "-1/2", "9/5" ], [ "-7/10", "8/5" ], [ "-1/2", "9/5" ], [ "-7/10", "8/5" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 55, "paper_system_label": "System 55", "local_name": "S055__DCB-LIT-P4-020" }, { "record_id": "DCB-LIT-P4-021", "problem": { "problem_id": "dcb_lit_p4_021", "task_type": "safety", "name": "DynaCertBench System 056 · DCB-LIT-P4-021 · ~\\texttt{S028}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = -0.4*x1+0.25*x2*x3-0.15*x1**(3), dx2/dt = -0.5*x2+0.3*x1*x3-0.2*x2**(3), dx3/dt = -0.6*x3+0.2*x1*x2-0.18*x1**(2)*x2", "source": "paper System 56; canonical record DCB-LIT-P4-021", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-021.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "0.5" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "3.5", "complement": true }, "analysis_bounds": [ [ "-9/2", "9/2" ], [ "-9/2", "9/2" ], [ "-9/2", "9/2" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 56, "paper_system_label": "System 56", "local_name": "S056__DCB-LIT-P4-021" }, { "record_id": "DCB-LIT-P4-022", "problem": { "problem_id": "dcb_lit_p4_022", "task_type": "safety", "name": "DynaCertBench System 057 · DCB-LIT-P4-022 · ~\\texttt{S031}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = -0.3*x1+0.2*x2*x4-0.15*x3**(2), dx2/dt = -0.4*x2+0.25*x1*x3-0.18*x2*x4, dx3/dt = -0.5*x3+0.3*x1*x4+0.12*x2**(2), dx4/dt = -0.6*x4+0.2*x2*x3-0.1*x1**(2)*x4", "source": "paper System 57; canonical record DCB-LIT-P4-022", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-022.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "ball", "center": [ "0", "0", "0", "0" ], "radius": "0.4" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0", "0" ], "radius": "3", "complement": true }, "analysis_bounds": [ [ "-4", "4" ], [ "-4", "4" ], [ "-4", "4" ], [ "-4", "4" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 57, "paper_system_label": "System 57", "local_name": "S057__DCB-LIT-P4-022" }, { "record_id": "DCB-LIT-P4-023", "problem": { "problem_id": "dcb_lit_p4_023", "task_type": "safety", "name": "DynaCertBench System 058 · DCB-LIT-P4-023 · ~\\texttt{S036}, \\cite{taheri2025barrierbench}", "dynamics": "x1[k+1] = 0.85*x1[k]+u0[k], x2[k+1] = 0.8*x2[k]+0.15*x1[k]+u1[k]", "source": "paper System 58; canonical record DCB-LIT-P4-023", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-023.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1", "initial_set": { "type": "box", "bounds": [ [ "1.5", "2" ], [ "1.2", "1.8" ] ] }, "unsafe_set": { "type": "union", "sets": [ { "type": "box", "bounds": [ [ "4.5", "5.5" ], [ "4", "5" ] ] }, { "type": "box", "bounds": [ [ "-5.5", "-4.5" ], [ "-5", "-4" ] ] } ] }, "analysis_bounds": [ [ "1/2", "3" ], [ "1/5", "14/5" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 58, "paper_system_label": "System 58", "local_name": "S058__DCB-LIT-P4-023" }, { "record_id": "DCB-LIT-P4-024", "problem": { "problem_id": "dcb_lit_p4_024", "task_type": "safety", "name": "DynaCertBench System 059 · DCB-LIT-P4-024 · ~\\texttt{S037}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = x2+u0, dx2/dt = -0.2*x1-0.1*(x1**(2)-1)*x2+u1", "source": "paper System 59; canonical record DCB-LIT-P4-024", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-024.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1", "initial_set": { "type": "ball", "center": [ "0", "0" ], "radius": "0.5" }, "unsafe_set": { "type": "ball", "center": [ "0", "0" ], "radius": "2.8", "complement": true }, "analysis_bounds": [ [ "-19/5", "19/5" ], [ "-19/5", "19/5" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 59, "paper_system_label": "System 59", "local_name": "S059__DCB-LIT-P4-024" }, { "record_id": "DCB-LIT-P4-025", "problem": { "problem_id": "dcb_lit_p4_025", "task_type": "safety", "name": "DynaCertBench System 060 · DCB-LIT-P4-025 · ~\\texttt{S042}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = x2+u0, dx2/dt = -0.6*x1-0.1*x2+0.05*x3+u1, dx3/dt = x4+u2, dx4/dt = -0.8*x3-0.15*x4+0.05*x1+u3", "source": "paper System 60; canonical record DCB-LIT-P4-025", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-025.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1,u2,u3", "initial_set": { "type": "ball", "center": [ "0", "0", "0", "0" ], "radius": "0.6" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0", "0" ], "radius": "3.5", "complement": true }, "analysis_bounds": [ [ "-9/2", "9/2" ], [ "-9/2", "9/2" ], [ "-9/2", "9/2" ], [ "-9/2", "9/2" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 60, "paper_system_label": "System 60", "local_name": "S060__DCB-LIT-P4-025" }, { "record_id": "DCB-LIT-P4-026", "problem": { "problem_id": "dcb_lit_p4_026", "task_type": "safety", "name": "DynaCertBench System 061 · DCB-LIT-P4-026 · ~\\texttt{S048}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = -0.5*x1+0.3*x2*sin(x3)-0.1*x1**(2)*x2, dx2/dt = -0.6*x2+0.5*x1*cos(x3)+0.15*x3**(2)*x1, dx3/dt = -0.7*x3-0.1*x3**(3)", "source": "paper System 61; canonical record DCB-LIT-P4-026", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-026.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "0.4" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "3.5", "complement": true }, "analysis_bounds": [ [ "-9/2", "9/2" ], [ "-9/2", "9/2" ], [ "-9/2", "9/2" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 61, "paper_system_label": "System 61", "local_name": "S061__DCB-LIT-P4-026" }, { "record_id": "DCB-LIT-P4-027", "problem": { "problem_id": "dcb_lit_p4_027", "task_type": "safety", "name": "DynaCertBench System 062 · DCB-LIT-P4-027 · ~\\texttt{S052}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = x2+u0, dx2/dt = x3+u1, dx3/dt = -6*x3-2.92*x2-1.2*x1+x1**(2)+u2", "source": "paper System 62; canonical record DCB-LIT-P4-027", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-027.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1,u2", "initial_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "0.1" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "2", "complement": true }, "analysis_bounds": [ [ "-3", "3" ], [ "-3", "3" ], [ "-3", "3" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 62, "paper_system_label": "System 62", "local_name": "S062__DCB-LIT-P4-027" }, { "record_id": "DCB-LIT-P4-028", "problem": { "problem_id": "dcb_lit_p4_028", "task_type": "safety", "name": "DynaCertBench System 063 · DCB-LIT-P4-028 · ~\\texttt{S057}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = -x1+0.08*x2+x1**(2)*x2+u0, dx2/dt = 0.6-0.08*x2-x1**(2)*x2+u1", "source": "paper System 63; canonical record DCB-LIT-P4-028", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-028.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1", "initial_set": { "type": "box", "bounds": [ [ "0.1", "0.5" ], [ "1", "2" ] ] }, "unsafe_set": { "type": "ball", "center": [ "0", "0" ], "radius": "5", "complement": true }, "analysis_bounds": [ [ "-6", "6" ], [ "-6", "6" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 63, "paper_system_label": "System 63", "local_name": "S063__DCB-LIT-P4-028" }, { "record_id": "DCB-LIT-P4-029", "problem": { "problem_id": "dcb_lit_p4_029", "task_type": "safety", "name": "DynaCertBench System 064 · DCB-LIT-P4-029 · ~\\texttt{S058}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = x2+u0, dx2/dt = -0.1*x1-0.05*x2+0.02*x1*x3+u1, dx3/dt = -0.2*x3+0.1*x1+u2", "source": "paper System 64; canonical record DCB-LIT-P4-029", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-029.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1,u2", "initial_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "0.5" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "3", "complement": true }, "analysis_bounds": [ [ "-4", "4" ], [ "-4", "4" ], [ "-4", "4" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 64, "paper_system_label": "System 64", "local_name": "S064__DCB-LIT-P4-029" }, { "record_id": "DCB-LIT-P4-030", "problem": { "problem_id": "dcb_lit_p4_030", "task_type": "safety", "name": "DynaCertBench System 065 · DCB-LIT-P4-030 · ~\\texttt{S059}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = -0.25*x1-x2**(2)-x3**(2)+0.75+u0, dx2/dt = -x2+x1*x3+u1, dx3/dt = -x3+x1*x2-0.1*x1+u2", "source": "paper System 65; canonical record DCB-LIT-P4-030", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-030.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1,u2", "initial_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "0.2" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "3", "complement": true }, "analysis_bounds": [ [ "-4", "4" ], [ "-4", "4" ], [ "-4", "4" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 65, "paper_system_label": "System 65", "local_name": "S065__DCB-LIT-P4-030" }, { "record_id": "DCB-LIT-P4-031", "problem": { "problem_id": "dcb_lit_p4_031", "task_type": "safety", "name": "DynaCertBench System 066 · DCB-LIT-P4-031 · ~\\texttt{S060}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = x1*(x1-0.1)*(1-x1)-0.3*x1*x2+u0, dx2/dt = -0.2*x2+0.1*x1*x2+u1", "source": "paper System 66; canonical record DCB-LIT-P4-031", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-031.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1", "initial_set": { "type": "box", "bounds": [ [ "0.2", "0.8" ], [ "0.1", "0.6" ] ] }, "unsafe_set": { "type": "box", "bounds": [ [ "1.5", "2" ], [ "1", "1.5" ] ] }, "analysis_bounds": [ [ "-4/5", "3" ], [ "-9/10", "5/2" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 66, "paper_system_label": "System 66", "local_name": "S066__DCB-LIT-P4-031" }, { "record_id": "DCB-LIT-P4-032", "problem": { "problem_id": "dcb_lit_p4_032", "task_type": "safety", "name": "DynaCertBench System 067 · DCB-LIT-P4-032 · ~\\texttt{S062}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = x2+u0, dx2/dt = -x1+x2*x3+u1, dx3/dt = 1-x2**(2)+u2", "source": "paper System 67; canonical record DCB-LIT-P4-032", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-032.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1,u2", "initial_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "0.2" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "2.5", "complement": true }, "analysis_bounds": [ [ "-7/2", "7/2" ], [ "-7/2", "7/2" ], [ "-7/2", "7/2" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 67, "paper_system_label": "System 67", "local_name": "S067__DCB-LIT-P4-032" }, { "record_id": "DCB-LIT-P4-033", "problem": { "problem_id": "dcb_lit_p4_033", "task_type": "safety", "name": "DynaCertBench System 068 · DCB-LIT-P4-033 · ~\\texttt{S076}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = x3+x2*x1+u0, dx2/dt = -x2-x1**(2)+u1, dx3/dt = -x1-0.5*x3+u2", "source": "paper System 68; canonical record DCB-LIT-P4-033", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-033.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1,u2", "initial_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "0.3" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "3.5", "complement": true }, "analysis_bounds": [ [ "-9/2", "9/2" ], [ "-9/2", "9/2" ], [ "-9/2", "9/2" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 68, "paper_system_label": "System 68", "local_name": "S068__DCB-LIT-P4-033" }, { "record_id": "DCB-LIT-P4-034", "problem": { "problem_id": "dcb_lit_p4_034", "task_type": "safety", "name": "DynaCertBench System 069 · DCB-LIT-P4-034 · ~\\texttt{S077}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = x2+u0, dx2/dt = -0.8*x1-0.3*x2*(1+x1**(2))+0.1*x3+u1, dx3/dt = x4+u2, dx4/dt = -0.9*x3-0.25*x4*(1+x3**(2))+0.1*x1+u3", "source": "paper System 69; canonical record DCB-LIT-P4-034", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-034.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1,u2,u3", "initial_set": { "type": "ball", "center": [ "0", "0", "0", "0" ], "radius": "0.35" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0", "0" ], "radius": "2.8", "complement": true }, "analysis_bounds": [ [ "-19/5", "19/5" ], [ "-19/5", "19/5" ], [ "-19/5", "19/5" ], [ "-19/5", "19/5" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 69, "paper_system_label": "System 69", "local_name": "S069__DCB-LIT-P4-034" }, { "record_id": "DCB-LIT-P4-035", "problem": { "problem_id": "dcb_lit_p4_035", "task_type": "safety", "name": "DynaCertBench System 070 · DCB-LIT-P4-035 · ~\\texttt{S087}, \\cite{taheri2025barrierbench}", "dynamics": "x1[k+1] = -1/100*x2, x2[k+1] = 1/100*x1", "formal_domain": "unbounded", "initial_set": { "type": "box", "bounds": [ [ "1/10", "2/5" ], [ "1/10", "11/20" ] ] }, "unsafe_set": { "type": "box", "bounds": [ [ "9/20", "1/2" ], [ "3/5", 1 ] ] }, "analysis_bounds": [ [ -2, 2 ], [ -2, 2 ] ], "source": "paper System 70; canonical record DCB-LIT-P4-035", "assumptions": [ "The source declares X = R^2, so all formal conditions are global.", "The displayed Euler map simplifies exactly to x1+ = -x2/100 and x2+ = x1/100.", "The finite analysis box is only a FastCheck screening box." ] }, "candidate": null, "paper_system_number": 70, "paper_system_label": "System 70", "local_name": "S070__DCB-LIT-P4-035" }, { "record_id": "DCB-LIT-P4-036", "problem": { "problem_id": "dcb_lit_p4_036", "task_type": "safety", "name": "DynaCertBench System 071 · DCB-LIT-P4-036 · ~\\texttt{U000}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = -0.5*x1+0.3*x2, dx2/dt = -0.4*x2+0.15*cos(x1)*sin(x3), dx3/dt = -0.6*x3+0.1*sin(x2)*cos(x4), dx4/dt = -0.5*x4", "source": "paper System 71; canonical record DCB-LIT-P4-036", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-036.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "ball", "center": [ "0", "0", "0", "0" ], "radius": "0.3" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0", "0" ], "radius": "2.5", "complement": true }, "analysis_bounds": [ [ "-7/2", "7/2" ], [ "-7/2", "7/2" ], [ "-7/2", "7/2" ], [ "-7/2", "7/2" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 71, "paper_system_label": "System 71", "local_name": "S071__DCB-LIT-P4-036" }, { "record_id": "DCB-LIT-P4-037", "problem": { "problem_id": "dcb_lit_p4_037", "task_type": "safety", "name": "DynaCertBench System 072 · DCB-LIT-P4-037 · ~\\texttt{U001}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = -0.3*x1+0.5*sin(x2), dx2/dt = -0.4*x2-0.6*sin(x1)+0.1*sin(x3), dx3/dt = -0.5*x3+0.2*sin(x1)*sin(x2)", "source": "paper System 72; canonical record DCB-LIT-P4-037", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-037.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "0.4" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "3", "complement": true }, "analysis_bounds": [ [ "-4", "4" ], [ "-4", "4" ], [ "-4", "4" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 72, "paper_system_label": "System 72", "local_name": "S072__DCB-LIT-P4-037" }, { "record_id": "DCB-LIT-P4-038", "problem": { "problem_id": "dcb_lit_p4_038", "task_type": "safety", "name": "DynaCertBench System 073 · DCB-LIT-P4-038 · ~\\texttt{U002}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = -0.3*x1+0.2*sin(x2)*exp(-x3**(2)), dx2/dt = -0.4*x2+0.15*cos(x1)*exp(-x1**(2)), dx3/dt = -0.5*x3+0.1*sin(x1+x2)", "source": "paper System 73; canonical record DCB-LIT-P4-038", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-038.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "0.45" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0" ], "radius": "3.2", "complement": true }, "analysis_bounds": [ [ "-21/5", "21/5" ], [ "-21/5", "21/5" ], [ "-21/5", "21/5" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 73, "paper_system_label": "System 73", "local_name": "S073__DCB-LIT-P4-038" }, { "record_id": "DCB-LIT-P4-039", "problem": { "problem_id": "dcb_lit_p4_039", "task_type": "safety", "name": "DynaCertBench System 074 · DCB-LIT-P4-039 · ~\\texttt{U003}, \\cite{taheri2025barrierbench}", "dynamics": "x1[k+1] = 4/5*x1 + 1/10*sin(x2), x2[k+1] = 17/20*x2 + 2/25*x1*x3, x3[k+1] = 9/10*x3 + 2/25*cos(x1*x2)", "formal_domain": "unbounded", "initial_set": { "type": "box", "bounds": [ [ "1/10", "2/5" ], [ "1/10", "3/10" ], [ "1/10", "2/5" ] ] }, "unsafe_set": { "type": "box", "bounds": [ [ 2, "5/2" ], [ "9/5", "11/5" ], [ "9/5", "23/10" ] ] }, "analysis_bounds": [ [ -3, 3 ], [ -3, 3 ], [ -3, 3 ] ], "source": "paper System 74; canonical record DCB-LIT-P4-039", "assumptions": [ "The source declares X = R^3, so all formal conditions are global.", "The finite analysis box is only a FastCheck screening box.", "BarrierBench reports no successful source certificate for U003." ] }, "candidate": null, "paper_system_number": 74, "paper_system_label": "System 74", "local_name": "S074__DCB-LIT-P4-039" }, { "record_id": "DCB-LIT-P4-040", "problem": { "problem_id": "dcb_lit_p4_040", "task_type": "safety", "name": "DynaCertBench System 075 · DCB-LIT-P4-040 · ~\\texttt{U004}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = -0.65*x1 + 0.1*x2**2*cos(x3), dx2/dt = -0.7*x2 + 0.12*sin(x1)*tanh(x4), dx3/dt = -0.75*x3 + 0.08*x1*x2, dx4/dt = -0.6*x4 + 0.06*(x1 + x2)", "initial_set": { "type": "box", "bounds": [ [ "1/5", "1/2" ], [ "-3/10", "0" ], [ "1/10", "2/5" ], [ "-1/5", "1/10" ] ] }, "unsafe_set": { "type": "box", "bounds": [ [ "-11/5", "-17/10" ], [ "8/5", "21/10" ], [ "-19/10", "-7/5" ], [ "13/10", "9/5" ] ] }, "analysis_bounds": [ [ -3, 3 ], [ -3, 3 ], [ -3, 3 ], [ -3, 3 ] ], "formal_domain": "unbounded", "source": "paper System 75; canonical record DCB-LIT-P4-040", "assumptions": [ "The catalogue states X = R^4; formal verification is global. The box [-3,3]^4 is used only for FastCheck screening.", "The initial set is [1/5,1/2] x [-3/10,0] x [1/10,2/5] x [-1/5,1/10].", "The unsafe set is [-11/5,-17/10] x [8/5,21/10] x [-19/10,-7/5] x [13/10,9/5]." ] }, "candidate": null, "paper_system_number": 75, "paper_system_label": "System 75", "local_name": "S075__DCB-LIT-P4-040" }, { "record_id": "DCB-LIT-P4-041", "problem": { "problem_id": "dcb_lit_p4_041", "task_type": "safety", "name": "DynaCertBench System 076 · DCB-LIT-P4-041 · ~\\texttt{U005}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = -0.58*x1+0.12*sin(x2)*cos(x3), dx2/dt = -0.62*x2+0.1*exp(-x1**(2))*tanh(x3), dx3/dt = -0.66*x3+0.08*sin(x1*x2)*exp(-x3**(2))", "source": "paper System 76; canonical record DCB-LIT-P4-041", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-041.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "box", "bounds": [ [ "-0.3", "0.2" ], [ "0.1", "0.4" ], [ "-0.2", "0.3" ] ] }, "unsafe_set": { "type": "box", "bounds": [ [ "-2.3", "-1.8" ], [ "1.7", "2.2" ], [ "-2", "-1.5" ] ] }, "analysis_bounds": [ [ "-33/10", "6/5" ], [ "-9/10", "16/5" ], [ "-3", "13/10" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 76, "paper_system_label": "System 76", "local_name": "S076__DCB-LIT-P4-041" }, { "record_id": "DCB-LIT-P4-042", "problem": { "problem_id": "dcb_lit_p4_042", "task_type": "safety", "name": "DynaCertBench System 077 · DCB-LIT-P4-042 · ~\\texttt{U006}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = u0*cos(x3), dx2/dt = u0*sin(x3), dx3/dt = u1", "source": "paper System 77; canonical record DCB-LIT-P4-042", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-042.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The unsafe set is the complement of the outer box, as specified by complement=true in the source JSON." ], "controller_parameters": "u0,u1", "initial_set": { "type": "box", "bounds": [ [ "-1", "1" ], [ "-1", "1" ], [ "-0.5", "0.5" ] ] }, "unsafe_set": { "type": "box", "bounds": [ [ "-5", "5" ], [ "-5", "5" ], [ "-3.14", "3.14" ] ], "complement": true }, "analysis_bounds": [ [ "-6", "6" ], [ "-6", "6" ], [ "-207/50", "207/50" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 77, "paper_system_label": "System 77", "local_name": "S077__DCB-LIT-P4-042" }, { "record_id": "DCB-LIT-P4-043", "problem": { "problem_id": "dcb_lit_p4_043", "task_type": "safety", "name": "DynaCertBench System 078 · DCB-LIT-P4-043 · ~\\texttt{U007}, \\cite{taheri2025barrierbench}", "dynamics": "x1[k+1] = 0.75*x1[k]+0.1*tanh(x2[k])*tanh(x3[k]), x2[k+1] = 0.82*x2[k]+0.09*sin(x1[k])*exp(-x3[k]**(2)), x3[k+1] = 0.79*x3[k]+0.08*cos(x1[k])*cos(x2[k])", "source": "paper System 78; canonical record DCB-LIT-P4-043", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-043.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "ball", "center": [ "1", "-0.5", "0.8" ], "radius": "0.5" }, "unsafe_set": { "type": "ball", "center": [ "-2.5", "2", "-2.2" ], "radius": "0.6" }, "analysis_bounds": [ [ "-41/10", "5/2" ], [ "-2", "18/5" ], [ "-19/5", "23/10" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 78, "paper_system_label": "System 78", "local_name": "S078__DCB-LIT-P4-043" }, { "record_id": "DCB-LIT-P4-044", "problem": { "problem_id": "dcb_lit_p4_044", "task_type": "safety", "name": "DynaCertBench System 079 · DCB-LIT-P4-044 · ~\\texttt{U008}, \\cite{taheri2025barrierbench}", "dynamics": "x1[k+1] = 0.73*x1[k]+0.13*sin(x2[k])*tanh(x3[k]), x2[k+1] = 0.89*x2[k]+0.17*cos(x1[k]**(2))*exp(-x3[k]**(2)), x3[k+1] = 0.64*x3[k]+0.22*tanh(x1[k]*x2[k])", "source": "paper System 79; canonical record DCB-LIT-P4-044", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-044.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "ball", "center": [ "0.9", "-0.7", "1.2" ], "radius": "0.55" }, "unsafe_set": { "type": "ball", "center": [ "-3.1", "2.8", "-2.6" ], "radius": "0.75" }, "analysis_bounds": [ [ "-97/20", "49/20" ], [ "-9/4", "91/20" ], [ "-87/20", "11/4" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 79, "paper_system_label": "System 79", "local_name": "S079__DCB-LIT-P4-044" }, { "record_id": "DCB-LIT-P4-045", "problem": { "problem_id": "dcb_lit_p4_045", "task_type": "safety", "name": "DynaCertBench System 080 · DCB-LIT-P4-045 · ~\\texttt{U009}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = -0.62*x1+0.1*sin(x2+x3), dx2/dt = -0.67*x2+0.08*cos(x1)*exp(-x3**(2)), dx3/dt = -0.64*x3+0.09*tanh(x1)*sin(x2)", "source": "paper System 80; canonical record DCB-LIT-P4-045", "assumptions": [ "Deterministically normalized from canonical record DCB-LIT-P4-045.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "initial_set": { "type": "box", "bounds": [ [ "-0.2", "0.3" ], [ "0.1", "0.5" ], [ "-0.3", "0.2" ] ] }, "unsafe_set": { "type": "box", "bounds": [ [ "2", "2.6" ], [ "-2.4", "-1.8" ], [ "2.1", "2.7" ] ] }, "analysis_bounds": [ [ "-6/5", "18/5" ], [ "-17/5", "3/2" ], [ "-13/10", "37/10" ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 80, "paper_system_label": "System 80", "local_name": "S080__DCB-LIT-P4-045" }, { "record_id": "DCB-EXT-P1-001", "problem": { "problem_id": "dcb_ext_p1_001", "task_type": "stability", "name": "DynaCertBench System 081 · DCB-EXT-P1-001 · Extend-Table~1 row~1 (NonPoly0), \\cite{abate2021fossil}", "dynamics": "dx1/dt = -x1+x1*x2, dx2/dt = -x2", "source": "paper System 81; canonical record DCB-EXT-P1-001", "assumptions": [ "Deterministically normalized from canonical record DCB-EXT-P1-001.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The target is global asymptotic stability; uniform Euclidean global exponential stability is not claimed." ], "equilibrium": [ 0, 0 ], "stability_kind": "global_asymptotic", "neighborhood": { "type": "unbounded" }, "analysis_bounds": [ [ -10, 10 ], [ -10, 10 ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 81, "paper_system_label": "System 81", "local_name": "S081__DCB-EXT-P1-001" }, { "record_id": "DCB-EXT-P1-002", "problem": { "problem_id": "dcb_ext_p1_002", "task_type": "stability", "name": "DynaCertBench System 082 · DCB-EXT-P1-002 · Extend-Table~1 row~6 (Poly2), \\cite{abate2021fossil}", "dynamics": "dx1/dt = -x1**3+x2, dx2/dt = -x1-x2", "source": "paper System 82; canonical record DCB-EXT-P1-002", "assumptions": [ "Deterministically normalized from canonical record DCB-EXT-P1-002.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "equilibrium": [ 0, 0 ], "stability_kind": "global_exponential", "neighborhood": { "type": "unbounded" }, "analysis_bounds": [ [ -10, 10 ], [ -10, 10 ] ], "formal_domain": "unbounded" }, "candidate": null, "paper_system_number": 82, "paper_system_label": "System 82", "local_name": "S082__DCB-EXT-P1-002" }, { "record_id": "DCB-EXT-P1-003", "problem": { "problem_id": "dcb_ext_p1_003", "task_type": "stability", "name": "DynaCertBench System 083 · DCB-EXT-P1-003 · Extend-Table~4 \\#2 (SineModel), \\cite{edwards2024fossil}", "dynamics": "dx1/dt = x2+u1, dx2/dt = u2+((3/20)*(981/100)*(1/2)*sin(x1)-(1/10)*x2)/((3/20)*(1/2)**2)", "source": "paper System 83; canonical record DCB-EXT-P1-003", "assumptions": [ "Deterministically normalized from canonical record DCB-EXT-P1-005.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u1,u2", "equilibrium": [ 0, 0 ], "stability_kind": "local_exponential", "neighborhood": { "type": "constraints", "constraints": [ { "lhs": "x1**2 + x2**2", "relation": "le", "rhs": "(1)**2" } ], "bounds": [ [ "-1", "1" ], [ "-1", "1" ] ] }, "analysis_bounds": [ [ "-1", "1" ], [ "-1", "1" ] ], "control_bounds": [ [ -5, 5 ], [ -5, 5 ] ] }, "candidate": null, "paper_system_number": 83, "paper_system_label": "System 83", "local_name": "S083__DCB-EXT-P1-003" }, { "record_id": "DCB-EXT-P2-001", "problem": { "problem_id": "dcb_ext_p2_001", "task_type": "roa", "name": "DynaCertBench System 084 · DCB-EXT-P2-001 · Extend-Section~5.2 and Appendix~C.3.2, \\cite{zhou2022neural}", "dynamics": "dx1/dt = sin(x2), dx2/dt = u-cos(x2)/(1-x1)", "controller_parameters": "u", "controller_observations": { "u": [ "x1", "x2" ] }, "control_bounds": [ [ -5, 5 ] ], "equilibrium": [ 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "ball", "center": [ 0, 0 ], "radius": "4/5" }, "analysis_bounds": [ [ "-4/5", "4/5" ], [ "-4/5", "4/5" ] ], "source": "paper System 84; canonical record DCB-EXT-P2-001", "assumptions": [ "The source controller and annular epsilon-ROA result are provenance only; this extension jointly synthesizes a new full-state feedback controller, Lyapunov function, and exact level c.", "The controller must remain in [-5,5] on the certified sublevel set Omega_c.", "The closed-loop equilibrium obligation is checked independently; for these dynamics it forces u(0,0)=1, not 0.", "No separate initial set is supplied; the target is to certify Omega_c itself inside x1^2+x2^2 <= (4/5)^2." ] }, "candidate": null, "paper_system_number": 84, "paper_system_label": "System 84", "local_name": "S084__DCB-EXT-P2-001" }, { "record_id": "DCB-EXT-P3-001", "problem": { "problem_id": "dcb_ext_p3_001", "task_type": "invariant", "name": "DynaCertBench System 085 · DCB-EXT-P3-001 · Extend--2D quadrotor, \\cite{meng2024physics}", "dynamics": "dx1/dt = x4, dx2/dt = x5, dx3/dt = x6, dx4/dt = 9.81*sin(x3)+sin(x3)*u1, dx5/dt = -9.81+9.81*cos(x3)+cos(x3)*u1, dx6/dt = u2", "source": "paper System 85; canonical record DCB-EXT-P3-001", "assumptions": [ "Deterministically normalized from canonical record DCB-EXT-P3-001.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u1,u2", "unsafe_set": { "type": "box", "bounds": [ [ "-(1/5)", "1/5" ], [ "-(1/5)", "1/5" ], [ "-(1/5)", "1/5" ], [ "-(1/5)", "1/5" ], [ "-(1/5)", "1/5" ], [ "-(1/5)", "1/5" ] ], "complement": true }, "analysis_bounds": [ [ "-1/5", "1/5" ], [ "-1/5", "1/5" ], [ "-1/5", "1/5" ], [ "-1/5", "1/5" ], [ "-1/5", "1/5" ], [ "-1/5", "1/5" ] ], "safety_interpretation": "outside_X_is_unsafe", "control_bounds": [ [ -5, 5 ], [ -5, 5 ] ] }, "candidate": null, "paper_system_number": 85, "paper_system_label": "System 85", "local_name": "S085__DCB-EXT-P3-001" }, { "record_id": "DCB-EXT-P3-002", "problem": { "problem_id": "dcb_ext_p3_002", "task_type": "invariant", "name": "DynaCertBench System 086 · DCB-EXT-P3-002 · Extend--3D quadrotor, \\cite{meng2024physics}", "dynamics": "dx1/dt = x4, dx2/dt = x5, dx3/dt = x6, dx4/dt = -9.81*sin(x8)-sin(x8)*u1, dx5/dt = 9.81*cos(x8)*sin(x7)+cos(x8)*sin(x7)*u1, dx6/dt = 9.81*cos(x8)*cos(x7)-9.81+cos(x8)*cos(x7)*u1, dx7/dt = u2, dx8/dt = u3, dx9/dt = u4", "source": "paper System 86; canonical record DCB-EXT-P3-002", "assumptions": [ "Deterministically normalized from canonical record DCB-EXT-P3-002.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u1,u2,u3,u4", "unsafe_set": { "type": "box", "bounds": [ [ "-(1/5)", "1/5" ], [ "-(1/5)", "1/5" ], [ "-(1/5)", "1/5" ], [ "-(1/5)", "1/5" ], [ "-(1/5)", "1/5" ], [ "-(1/5)", "1/5" ], [ "-(1/5)", "1/5" ], [ "-(1/5)", "1/5" ], [ "-(1/5)", "1/5" ] ], "complement": true }, "analysis_bounds": [ [ "-1/5", "1/5" ], [ "-1/5", "1/5" ], [ "-1/5", "1/5" ], [ "-1/5", "1/5" ], [ "-1/5", "1/5" ], [ "-1/5", "1/5" ], [ "-1/5", "1/5" ], [ "-1/5", "1/5" ], [ "-1/5", "1/5" ] ], "safety_interpretation": "outside_X_is_unsafe", "control_bounds": [ [ -5, 5 ], [ -5, 5 ], [ -5, 5 ], [ -5, 5 ] ] }, "candidate": null, "paper_system_number": 86, "paper_system_label": "System 86", "local_name": "S086__DCB-EXT-P3-002" }, { "record_id": "DCB-EXT-P4-001", "problem": { "problem_id": "dcb_ext_p4_001", "task_type": "safety", "name": "DynaCertBench System 087 · DCB-EXT-P4-001 · Extend-\\texttt{S002}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = x2+u0, dx2/dt = -0.95*sin(x1)-0.02*x2+0.01*x3+u1, dx3/dt = -0.1*x3+0.02*x1+u2, dx4/dt = -0.03*x4+0.01*x2+u3", "source": "paper System 87; canonical record DCB-EXT-P4-001", "assumptions": [ "Deterministically normalized from canonical record DCB-EXT-P4-001.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1,u2,u3", "initial_set": { "type": "ball", "center": [ "0", "0", "0", "0" ], "radius": "0.3" }, "unsafe_set": { "type": "ball", "center": [ "0", "0", "0", "0" ], "radius": "2.5", "complement": true }, "analysis_bounds": [ [ "-7/2", "7/2" ], [ "-7/2", "7/2" ], [ "-7/2", "7/2" ], [ "-7/2", "7/2" ] ], "formal_domain": "unbounded", "control_bounds": [ [ -5, 5 ], [ -5, 5 ], [ -5, 5 ], [ -5, 5 ] ] }, "candidate": null, "paper_system_number": 87, "paper_system_label": "System 87", "local_name": "S087__DCB-EXT-P4-001" }, { "record_id": "DCB-EXT-P4-002", "problem": { "problem_id": "dcb_ext_p4_002", "task_type": "safety", "name": "DynaCertBench System 088 · DCB-EXT-P4-002 · Extend-\\texttt{S011}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = x2+u0, dx2/dt = -0.3*x1-0.2*(1+cos(x1))*x2-0.1*x1**(3)+u1", "source": "paper System 88; canonical record DCB-EXT-P4-002", "assumptions": [ "Deterministically normalized from canonical record DCB-EXT-P4-002.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1", "initial_set": { "type": "ball", "center": [ "0", "0" ], "radius": "0.6" }, "unsafe_set": { "type": "ball", "center": [ "2", "2" ], "radius": "1" }, "analysis_bounds": [ [ "-8/5", "4" ], [ "-8/5", "4" ] ], "formal_domain": "unbounded", "control_bounds": [ [ -1, 1 ], [ -1, 1 ] ] }, "candidate": null, "paper_system_number": 88, "paper_system_label": "System 88", "local_name": "S088__DCB-EXT-P4-002" }, { "record_id": "DCB-EXT-P4-003", "problem": { "problem_id": "dcb_ext_p4_003", "task_type": "safety", "name": "DynaCertBench System 089 · DCB-EXT-P4-003 · Extend-\\texttt{S024}, \\cite{taheri2025barrierbench}", "dynamics": "dx1/dt = x2+u0, dx2/dt = -x1-0.4*x2+0.15*sin(x3)+u1, dx3/dt = x4+u2, dx4/dt = -0.6*x3-0.3*x4+u3", "source": "paper System 89; canonical record DCB-EXT-P4-003", "assumptions": [ "Deterministically normalized from canonical record DCB-EXT-P4-003.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "controller_parameters": "u0,u1,u2,u3", "initial_set": { "type": "box", "bounds": [ [ "0.5", "0.8" ], [ "0.3", "0.6" ], [ "0.5", "0.8" ], [ "0.3", "0.6" ] ] }, "unsafe_set": { "type": "union", "sets": [ { "type": "box", "bounds": [ [ "2.5", "3.5" ], [ "2", "3" ], [ "2.5", "3.5" ], [ "2", "3" ] ] }, { "type": "box", "bounds": [ [ "-3.5", "-2.5" ], [ "-3", "-2" ], [ "-3.5", "-2.5" ], [ "-3", "-2" ] ] } ] }, "analysis_bounds": [ [ "-1/2", "9/5" ], [ "-7/10", "8/5" ], [ "-1/2", "9/5" ], [ "-7/10", "8/5" ] ], "formal_domain": "unbounded", "control_bounds": [ [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ] ] }, "candidate": null, "paper_system_number": 89, "paper_system_label": "System 89", "local_name": "S089__DCB-EXT-P4-003" }, { "record_id": "DCB-EXT-P4-004", "paper_system_number": 90, "paper_system_label": "System 90", "local_name": "S090__DCB-EXT-P4-004", "problem": { "problem_id": "dcb_ext_p4_004", "task_type": "safety", "name": "DynaCertBench System 090 · DCB-EXT-P4-004 · Extend--Hi-Ord8 with bounded sinusoidal disturbance, \\cite{abate2021fossil}", "dynamics": "dx1/dt = x2, dx2/dt = x3, dx3/dt = x4, dx4/dt = x5, dx5/dt = x6, dx6/dt = x7, dx7/dt = x8, dx8/dt = -20*x8+170*x7+800*x6+2273*x5+3980*x4+4180*x3+2400*x2+576*x1+sin(omega*x1)", "source": "paper System 90; canonical record DCB-EXT-P4-004", "assumptions": [ "Transcribed from paper-4-20-20260904 System 93.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "Safety is required for every fixed omega in [1/2,2].", "The synthesized barrier is one common state-only function B(x), independent of omega.", "PDF 4-20 removes d(t) and corrects the signs in the last flow component." ], "analysis_bounds": [ [ "-11/5", "11/5" ], [ "-11/5", "11/5" ], [ "-11/5", "11/5" ], [ "-11/5", "11/5" ], [ "-11/5", "11/5" ], [ "-11/5", "11/5" ], [ "-11/5", "11/5" ], [ "-11/5", "11/5" ] ], "fixed_parameters": [ { "name": "omega", "bounds": [ "1/2", 2 ] } ], "initial_set": { "type": "box", "bounds": [ [ "9/10", "11/10" ], [ "9/10", "11/10" ], [ "9/10", "11/10" ], [ "9/10", "11/10" ], [ "9/10", "11/10" ], [ "9/10", "11/10" ], [ "9/10", "11/10" ], [ "9/10", "11/10" ] ] }, "unsafe_set": { "type": "box", "bounds": [ [ "-11/5", "-9/5" ], [ "-11/5", "-9/5" ], [ "-11/5", "-9/5" ], [ "-11/5", "-9/5" ], [ "-11/5", "-9/5" ], [ "-11/5", "-9/5" ], [ "-11/5", "-9/5" ], [ "-11/5", "-9/5" ] ] } }, "candidate": null }, { "record_id": "DCB-EXT-P4-005", "paper_system_number": 91, "paper_system_label": "System 91", "local_name": "S091__DCB-EXT-P4-005", "problem": { "problem_id": "dcb_ext_p4_005", "task_type": "safety", "name": "DynaCertBench System 091 · DCB-EXT-P4-005 · Safety extension of Example~5.4, \\cite{liu2024tool}", "dynamics": "dx1/dt = -x1+1/2*x2-1/10*x9**2+sin(x2), dx2/dt = -1/2*x1-x2, dx3/dt = -x3+1/2*x4-1/10*x1**2+exp(-x3)-1, dx4/dt = -1/2*x3-x4, dx5/dt = -x5+1/2*x6+1/10*x7**2, dx6/dt = -1/2*x5-x6, dx7/dt = -x7+1/2*x8, dx8/dt = -1/2*x7-x8, dx9/dt = -x9+1/2*x10, dx10/dt = -1/2*x9-x10+1/10*x2**2", "source": "paper System 91; canonical record DCB-EXT-P4-005", "assumptions": [ "Transcribed from paper-4-20-20260904 System 94.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "PDF 4-20 removes omega and d(t); both perturbations are deterministic state functions." ], "analysis_bounds": [ [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ] ], "initial_set": { "type": "box", "bounds": [ [ "-7/20", "7/20" ], [ "-7/20", "7/20" ], [ "-7/20", "7/20" ], [ "-7/20", "7/20" ], [ "-7/20", "7/20" ], [ "-7/20", "7/20" ], [ "-7/20", "7/20" ], [ "-7/20", "7/20" ], [ "-7/20", "7/20" ], [ "-7/20", "7/20" ] ] }, "unsafe_set": { "type": "box", "bounds": [ [ "9/20", "4/5" ], [ "-1/5", "1/5" ], [ "-2/5", "2/5" ], [ "-2/5", "2/5" ], [ "-2/5", "2/5" ], [ "-2/5", "2/5" ], [ "-2/5", "2/5" ], [ "-2/5", "2/5" ], [ "-2/5", "2/5" ], [ "-2/5", "2/5" ] ] } }, "candidate": null }, { "record_id": "DCB-OURS-P1-001", "paper_system_number": 92, "paper_system_label": "System 92", "local_name": "S092__DCB-OURS-P1-001", "problem": { "problem_id": "dcb_ours_p1_001", "task_type": "stability", "name": "DynaCertBench System 092 · DCB-OURS-P1-001 · 8D Rational-Cosine Ring", "dynamics": "dx1/dt = -3*x1+x2*cos(x1)/(1+x8**2), dx2/dt = -3*x2+x3*cos(x2)/(1+x1**2), dx3/dt = -3*x3+x4*cos(x3)/(1+x2**2), dx4/dt = -3*x4+x5*cos(x4)/(1+x3**2), dx5/dt = -3*x5+x6*cos(x1)/(1+x4**2), dx6/dt = -3*x6+x7*cos(x2)/(1+x5**2), dx7/dt = -3*x7+x8*cos(x3)/(1+x6**2), dx8/dt = -3*x8+x1*cos(x4)/(1+x7**2)", "source": "paper System 92; canonical record DCB-OURS-P1-001", "assumptions": [ "Transcribed from active LaTeX System 92.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "analysis_bounds": [ [ -3, 3 ], [ -3, 3 ], [ -3, 3 ], [ -3, 3 ], [ -3, 3 ], [ -3, 3 ], [ -3, 3 ], [ -3, 3 ] ], "equilibrium": [ 0, 0, 0, 0, 0, 0, 0, 0 ], "stability_kind": "global_exponential", "neighborhood": { "type": "unbounded" } }, "candidate": null }, { "record_id": "DCB-OURS-P1-002", "paper_system_number": 93, "paper_system_label": "System 93", "local_name": "S093__DCB-OURS-P1-002", "problem": { "problem_id": "dcb_ours_p1_002", "task_type": "stability", "name": "DynaCertBench System 093 · DCB-OURS-P1-002 · 7D Heterogeneous Trigonometric-Rational Network", "dynamics": "dx1/dt = -5*x1+x2*cos(x1)/(1+x7**2), dx2/dt = -(9/2)*x2+(6/5)*x4*sin(2*x2)/(1+2*x1**2), dx3/dt = -6*x3+(4/5)*x6*cos(x3/2)/(1+(3/2)*x2**2), dx4/dt = -(11/2)*x4+(3/2)*x7*sin((3/2)*x4)/(1+3*x3**2), dx5/dt = -5*x5+(9/10)*x1*cos(2*x5)/(1+(4/3)*x4**2), dx6/dt = -(9/2)*x6+(7/5)*x5*sin(x6/3)/(1+(5/2)*x5**2), dx7/dt = -6*x7+(1/2)*x3*cos((5/2)*x7)/(1+(1/2)*x5**2)", "source": "paper System 93; canonical record DCB-OURS-P1-002", "assumptions": [ "Transcribed from active LaTeX System 93.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "analysis_bounds": [ [ -3, 3 ], [ -3, 3 ], [ -3, 3 ], [ -3, 3 ], [ -3, 3 ], [ -3, 3 ], [ -3, 3 ] ], "equilibrium": [ 0, 0, 0, 0, 0, 0, 0 ], "stability_kind": "global_exponential", "neighborhood": { "type": "unbounded" } }, "candidate": null }, { "record_id": "DCB-OURS-P2-001", "paper_system_number": 94, "paper_system_label": "System 94", "local_name": "S094__DCB-OURS-P2-001", "problem": { "problem_id": "dcb_ours_p2_001", "task_type": "roa", "name": "DynaCertBench System 094 · DCB-OURS-P2-001 · Our cyclic ten-dimensional rational system", "dynamics": "dx1/dt = -x1+1/4*x2+1/20*(x3/(1+x3**2)), dx2/dt = -x2+1/4*x3+1/20*(x4/(1+x4**2)), dx3/dt = -x3+1/4*x4+1/20*(x5/(1+x5**2)), dx4/dt = -x4+1/4*x5+1/20*(x6/(1+x6**2)), dx5/dt = -x5+1/4*x6+1/20*(x7/(1+x7**2)), dx6/dt = -x6+1/4*x7+1/20*(x8/(1+x8**2)), dx7/dt = -x7+1/4*x8+1/20*(x9/(1+x9**2)), dx8/dt = -x8+1/4*x9+1/20*(x10/(1+x10**2)), dx9/dt = -x9+1/4*x10+1/20*(x1/(1+x1**2)), dx10/dt = -x10+1/4*x1+1/20*(x2/(1+x2**2))", "source": "paper System 94; canonical record DCB-OURS-P2-001", "assumptions": [ "Transcribed from paper-4-20-20260904 System 96.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The formal neighborhood is R^10; [-10,10]^10 is numerical screening only." ], "analysis_bounds": [ [ -10, 10 ], [ -10, 10 ], [ -10, 10 ], [ -10, 10 ], [ -10, 10 ], [ -10, 10 ], [ -10, 10 ], [ -10, 10 ], [ -10, 10 ], [ -10, 10 ] ], "equilibrium": [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], "stability_kind": "local_asymptotic", "neighborhood": { "type": "unbounded" }, "formal_domain": "unbounded" }, "candidate": null }, { "record_id": "DCB-OURS-P2-002", "paper_system_number": 95, "paper_system_label": "System 95", "local_name": "S095__DCB-OURS-P2-002", "problem": { "problem_id": "dcb_ours_p2_002", "task_type": "roa", "name": "DynaCertBench System 095 · DCB-OURS-P2-002 · Seven-dimensional multi-machine swing system", "dynamics": "dx1/dt = x2, dx2/dt = -(1/2)*x2-(3/2)*sin(x1)-sin(x1-x3)-(3/2)*sin(x1-x7), dx3/dt = x4, dx4/dt = -(1/2)*x4-(1/2)*sin(x3)-sin(x3-x1)-sin(x3-x5), dx5/dt = x6, dx6/dt = -(1/2)*x6-sin(x5)-sin(x5-x3)-(3/2)*sin(x5-x7), dx7/dt = -(1/5)*sin(x7)-(3/2)*sin(x7-x1)-(3/2)*sin(x7-x5)", "source": "paper System 95; canonical record DCB-OURS-P2-002", "assumptions": [ "Transcribed from active LaTeX System 95.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "analysis_bounds": [ [ -3, 3 ], [ -3, 3 ], [ -3, 3 ], [ -3, 3 ], [ -3, 3 ], [ -3, 3 ], [ -3, 3 ] ], "equilibrium": [ 0, 0, 0, 0, 0, 0, 0 ], "neighborhood": { "type": "box", "bounds": [ [ "-pi", "pi" ], [ "-pi", "pi" ], [ "-pi", "pi" ], [ "-pi", "pi" ], [ "-pi", "pi" ], [ "-pi", "pi" ], [ "-pi", "pi" ] ] }, "initial_set": { "type": "box", "bounds": [ [ "-3/5", "3/5" ], [ "-3/5", "3/5" ], [ "-3/5", "3/5" ], [ "-3/5", "3/5" ], [ "-3/5", "3/5" ], [ "-3/5", "3/5" ], [ "-3/5", "3/5" ] ] }, "stability_kind": "local_asymptotic" }, "candidate": null }, { "record_id": "DCB-OURS-P2-003", "paper_system_number": 96, "paper_system_label": "System 96", "local_name": "S096__DCB-OURS-P2-003", "problem": { "problem_id": "dcb_ours_p2_003", "task_type": "roa", "name": "DynaCertBench System 096 · DCB-OURS-P2-003 · Seven-dimensional coupled saturating oscillator", "dynamics": "dx1/dt = x2, dx2/dt = -2*x1/(1+x1**2)+(1/2)*x3-(3/10)*x2+(1/5)*x5, dx3/dt = x4, dx4/dt = (1/2)*x1-(3/2)*x3-sin(x3)-(1/5)*x4+(1/10)*x6, dx5/dt = -(1/2)*x5-(1/5)*x2+(1/10)*x6, dx6/dt = -(2/5)*x6-(1/10)*x5-(1/10)*x4+(1/10)*x7, dx7/dt = -(3/10)*x7-(1/10)*x6", "source": "paper System 96; canonical record DCB-OURS-P2-003", "assumptions": [ "Transcribed from active LaTeX System 96.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "analysis_bounds": [ [ -5, 5 ], [ -5, 5 ], [ -5, 5 ], [ -5, 5 ], [ -5, 5 ], [ -5, 5 ], [ -5, 5 ] ], "equilibrium": [ 0, 0, 0, 0, 0, 0, 0 ], "neighborhood": { "type": "box", "bounds": [ [ "-5", "5" ], [ "-5", "5" ], [ "-5", "5" ], [ "-5", "5" ], [ "-5", "5" ], [ "-5", "5" ], [ "-5", "5" ] ] }, "initial_set": { "type": "box", "bounds": [ [ "-3/5", "3/5" ], [ "-3/5", "3/5" ], [ "-3/5", "3/5" ], [ "-3/5", "3/5" ], [ "-3/5", "3/5" ], [ "-3/5", "3/5" ], [ "-3/5", "3/5" ] ] }, "stability_kind": "local_asymptotic" }, "candidate": null }, { "record_id": "DCB-OURS-P3-001", "paper_system_number": 97, "paper_system_label": "System 97", "local_name": "S097__DCB-OURS-P3-001", "problem": { "problem_id": "dcb_ours_p3_001", "task_type": "invariant", "name": "DynaCertBench System 097 · DCB-OURS-P3-001 · New-6D Sin Cascade Invariance", "dynamics": "dx1/dt = x2+u0, dx2/dt = -x1-1/10*sin(x1)-x2+x3, dx3/dt = -2*x3+x4, dx4/dt = -3/2*x4+x5, dx5/dt = -2*x5+x6, dx6/dt = -3/2*x6+u1", "source": "paper System 97; canonical record DCB-OURS-P3-001", "assumptions": [ "Transcribed from paper-4-20-20260904 System 97.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "X_seed is the paper's required nontrivial seed set." ], "analysis_bounds": [ [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ] ], "controller_parameters": "u0,u1", "control_bounds": [ [ -1, 1 ], [ -1, 1 ] ], "initial_set": { "type": "ball", "center": [ 0, 0, 0, 0, 0, 0 ], "radius": "1/2" }, "unsafe_set": { "type": "ball", "center": [ 0, "4/5", 0, 0, 0, 0 ], "radius": "1/10" } }, "candidate": null }, { "record_id": "DCB-OURS-P3-002", "paper_system_number": 98, "paper_system_label": "System 98", "local_name": "S098__DCB-OURS-P3-002", "problem": { "problem_id": "dcb_ours_p3_002", "task_type": "invariant", "name": "DynaCertBench System 098 · DCB-OURS-P3-002 · New-6D Exponential Cascade Invariance", "dynamics": "dx1/dt = x2+u, dx2/dt = -x1-1/20*(exp(x1)-exp(-x1))-x2+x3, dx3/dt = -2*x3+x4, dx4/dt = -3/2*x4+x5, dx5/dt = -2*x5+x6, dx6/dt = -3/2*x6", "source": "paper System 98; canonical record DCB-OURS-P3-002", "assumptions": [ "Transcribed from paper-4-20-20260904 System 98.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The source X_I ball has radius 1/8." ], "analysis_bounds": [ [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ] ], "controller_parameters": "u", "control_bounds": [ [ -1, 1 ] ], "initial_set": { "type": "ball", "center": [ "-1/4", "1/4", 0, 0, 0, 0 ], "radius": "1/8" }, "unsafe_set": { "type": "ball", "center": [ "7/10", "-7/10", 0, 0, 0, 0 ], "radius": "1/10" } }, "candidate": null }, { "record_id": "DCB-OURS-P3-003", "paper_system_number": 99, "paper_system_label": "System 99", "local_name": "S099__DCB-OURS-P3-003", "problem": { "problem_id": "dcb_ours_p3_003", "task_type": "invariant", "name": "DynaCertBench System 099 · DCB-OURS-P3-003 · New-5D Mixed Nonlinear Cascade Invariance", "dynamics": "dx1/dt = x2-1/2*x1, dx2/dt = -x1-x1/(5*(1+x1**2))-2*x2+x3/(10*(1+x3**2)), dx3/dt = x4-1/2*x3, dx4/dt = -x3-1/40*(exp(x3)-exp(-x3))-2*x4+1/5*x5, dx5/dt = -3/2*x5-1/10*x5**3+u", "source": "paper System 99; canonical record DCB-OURS-P3-003", "assumptions": [ "Transcribed from paper-4-20-20260904 System 99.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "The paper supplies no X_I/X_seed; no seed set is invented locally." ], "analysis_bounds": [ [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ] ], "controller_parameters": "u", "control_bounds": [ [ -1, 1 ] ], "unsafe_set": { "type": "ball", "center": [ "7/10", "-7/10", 0, 0, 0 ], "radius": "1/10" } }, "candidate": null }, { "record_id": "DCB-OURS-P3-004", "paper_system_number": 100, "paper_system_label": "System 100", "local_name": "S100__DCB-OURS-P3-004", "problem": { "problem_id": "dcb_ours_p3_004", "task_type": "invariant", "name": "DynaCertBench System 100 · DCB-OURS-P3-004 · New-6D Opinion Dynamics", "dynamics": "x1[k+1] = x1+1/4*(tanh(x2-x1) + tanh(x3-x1)), x2[k+1] = x2+1/4*(tanh(x1-x2) + tanh(x3-x2) + tanh(x4-x2)), x3[k+1] = x3+1/4*(tanh(x1-x3) + tanh(x2-x3) + tanh(x5-x3)), x4[k+1] = x4+1/4*(tanh(x2-x4) + tanh(x5-x4)), x5[k+1] = x5+1/4*(tanh(x3-x5) + tanh(x4-x5) + tanh(x6-x5)), x6[k+1] = x6+1/4*(tanh(x5-x6))", "source": "paper System 100; canonical record DCB-OURS-P3-004", "assumptions": [ "Transcribed from paper-4-20-20260904 System 100.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "PDF 4-20 fixes N=6, alpha=1/4, and the displayed adjacency matrix.", "The paper supplies no X_I/X_seed; no seed set is invented locally." ], "analysis_bounds": [ [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ] ], "unsafe_set": { "type": "union", "sets": [ { "type": "constraint", "lhs": "x1", "relation": "ge", "rhs": "4/5" }, { "type": "constraint", "lhs": "x1", "relation": "le", "rhs": "-4/5" }, { "type": "constraint", "lhs": "x2", "relation": "ge", "rhs": "4/5" }, { "type": "constraint", "lhs": "x2", "relation": "le", "rhs": "-4/5" }, { "type": "constraint", "lhs": "x3", "relation": "ge", "rhs": "4/5" }, { "type": "constraint", "lhs": "x3", "relation": "le", "rhs": "-4/5" }, { "type": "constraint", "lhs": "x4", "relation": "ge", "rhs": "4/5" }, { "type": "constraint", "lhs": "x4", "relation": "le", "rhs": "-4/5" }, { "type": "constraint", "lhs": "x5", "relation": "ge", "rhs": "4/5" }, { "type": "constraint", "lhs": "x5", "relation": "le", "rhs": "-4/5" }, { "type": "constraint", "lhs": "x6", "relation": "ge", "rhs": "4/5" }, { "type": "constraint", "lhs": "x6", "relation": "le", "rhs": "-4/5" } ] } }, "candidate": null }, { "record_id": "DCB-OURS-P3-005", "paper_system_number": 101, "paper_system_label": "System 101", "local_name": "S101__DCB-OURS-P3-005", "problem": { "problem_id": "dcb_ours_p3_005", "task_type": "invariant", "name": "DynaCertBench System 101 · DCB-OURS-P3-005 · Invariance Verification of a Discrete-Time Controlled Hopfield Network", "dynamics": "x1[k+1] = x1+1/20*(1-x1**2)*(-1/2*log(1+x1)+1/2*log(1-x1)+-3/2*x1 + 1/4*x2+u1), x2[k+1] = x2+1/20*(1-x2**2)*(-1/2*log(1+x2)+1/2*log(1-x2)+-3/2*x2 + 1/4*x1 + 1/4*x3+u2), x3[k+1] = x3+1/20*(1-x3**2)*(-1/2*log(1+x3)+1/2*log(1-x3)+-3/2*x3 + 1/4*x2 + 1/4*x4+u3), x4[k+1] = x4+1/20*(1-x4**2)*(-1/2*log(1+x4)+1/2*log(1-x4)+-3/2*x4 + 1/4*x3 + 1/4*x5+u4), x5[k+1] = x5+1/20*(1-x5**2)*(-1/2*log(1+x5)+1/2*log(1-x5)+-3/2*x5 + 1/4*x4 + 1/4*x6+u5), x6[k+1] = x6+1/20*(1-x6**2)*(-1/2*log(1+x6)+1/2*log(1-x6)+-3/2*x6 + 1/4*x5 + 1/4*x7+u6), x7[k+1] = x7+1/20*(1-x7**2)*(-1/2*log(1+x7)+1/2*log(1-x7)+-3/2*x7 + 1/4*x6 + 1/4*x8+u7), x8[k+1] = x8+1/20*(1-x8**2)*(-1/2*log(1+x8)+1/2*log(1-x8)+-3/2*x8 + 1/4*x7 + 1/4*x9+u8), x9[k+1] = x9+1/20*(1-x9**2)*(-1/2*log(1+x9)+1/2*log(1-x9)+-3/2*x9 + 1/4*x8 + 1/4*x10+u9), x10[k+1] = x10+1/20*(1-x10**2)*(-1/2*log(1+x10)+1/2*log(1-x10)+-3/2*x10 + 1/4*x9+u10)", "source": "paper System 101; canonical record DCB-OURS-P3-005", "assumptions": [ "Transcribed from paper-4-20-20260904 System 101.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.", "atanh(x) is represented exactly by (log(1+x)-log(1-x))/2 on X.", "W is the tridiagonal matrix displayed in PDF 4-20: diagonal -3/2 and adjacent entries 1/4.", "The paper supplies no X_I/X_seed; no seed set is invented locally." ], "analysis_bounds": [ [ "-9/10", "9/10" ], [ "-9/10", "9/10" ], [ "-9/10", "9/10" ], [ "-9/10", "9/10" ], [ "-9/10", "9/10" ], [ "-9/10", "9/10" ], [ "-9/10", "9/10" ], [ "-9/10", "9/10" ], [ "-9/10", "9/10" ], [ "-9/10", "9/10" ] ], "controller_parameters": "u1,u2,u3,u4,u5,u6,u7,u8,u9,u10", "control_bounds": [ [ -2, 2 ], [ -2, 2 ], [ -2, 2 ], [ -2, 2 ], [ -2, 2 ], [ -2, 2 ], [ -2, 2 ], [ -2, 2 ], [ -2, 2 ], [ -2, 2 ] ], "safety_interpretation": "outside_X_is_unsafe" }, "candidate": null }, { "record_id": "DCB-OURS-P4-001", "paper_system_number": 102, "paper_system_label": "System 102", "local_name": "S102__DCB-OURS-P4-001", "problem": { "problem_id": "dcb_ours_p4_001", "task_type": "safety", "name": "DynaCertBench System 102 · DCB-OURS-P4-001 · New-8D Chain Safety", "dynamics": "x1[k+1] = x1+1/200*(-x1/(1+x1**2)-4*x2), x2[k+1] = x2+1/200*(4*x1/(1+x1**2)-x2-3*x3/(1+x3**2)), x3[k+1] = x3+1/200*(3*x2-x3/(1+x3**2)-5*x4), x4[k+1] = x4+1/200*(5*x3/(1+x3**2)-x4-2*x5/(1+x5**2)), x5[k+1] = x5+1/200*(2*x4-x5/(1+x5**2)-4*x6), x6[k+1] = x6+1/200*(4*x5/(1+x5**2)-x6-3*x7/(1+x7**2)), x7[k+1] = x7+1/200*(3*x6-x7/(1+x7**2)-6*x8), x8[k+1] = x8+1/200*(6*x7/(1+x7**2)-x8)", "source": "paper System 102; canonical record DCB-OURS-P4-001", "assumptions": [ "Transcribed from active LaTeX System 102.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "analysis_bounds": [ [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ] ], "initial_set": { "type": "box", "bounds": [ [ "-3/20", "3/20" ], [ "-3/20", "3/20" ], [ "-3/20", "3/20" ], [ "-3/20", "3/20" ], [ "-3/20", "3/20" ], [ "-3/20", "3/20" ], [ "-3/20", "3/20" ], [ "-3/20", "3/20" ] ] }, "unsafe_set": { "type": "box", "bounds": [ [ "3/4", "9/10" ], [ "-1/10", "1/10" ], [ "-1/10", "1/10" ], [ "-1/10", "1/10" ], [ "-1/10", "1/10" ], [ "-1/10", "1/10" ], [ "-1/10", "1/10" ], [ "-1/10", "1/10" ] ] } }, "candidate": null }, { "record_id": "DCB-OURS-P4-002", "paper_system_number": 103, "paper_system_label": "System 103", "local_name": "S103__DCB-OURS-P4-002", "problem": { "problem_id": "dcb_ours_p4_002", "task_type": "safety", "name": "DynaCertBench System 103 · DCB-OURS-P4-002 · New-6D Arctan Chain Safety", "dynamics": "x1[k+1] = x1+1/200*(-atan(x1)-8*x2), x2[k+1] = x2+1/200*(8*atan(x1)-x2-6*atan(x3)), x3[k+1] = x3+1/200*(6*x2-atan(x3)-10*x4), x4[k+1] = x4+1/200*(10*atan(x3)-x4-4*atan(x5)), x5[k+1] = x5+1/200*(4*x4-atan(x5)-8*x6), x6[k+1] = x6+1/200*(8*atan(x5)-x6)", "source": "paper System 103; canonical record DCB-OURS-P4-002", "assumptions": [ "Transcribed from paper-4-20-20260904 System 103.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "analysis_bounds": [ [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ] ], "initial_set": { "type": "box", "bounds": [ [ "-3/20", "3/20" ], [ "-3/20", "3/20" ], [ "-3/20", "3/20" ], [ "-3/20", "3/20" ], [ "-3/20", "3/20" ], [ "-3/20", "3/20" ] ] }, "unsafe_set": { "type": "box", "bounds": [ [ "3/4", "9/10" ], [ "-1/10", "1/10" ], [ "-1/10", "1/10" ], [ "-1/10", "1/10" ], [ "-1/10", "1/10" ], [ "-1/10", "1/10" ] ] } }, "candidate": null }, { "record_id": "DCB-OURS-P4-003", "paper_system_number": 104, "paper_system_label": "System 104", "local_name": "S104__DCB-OURS-P4-003", "problem": { "problem_id": "dcb_ours_p4_003", "task_type": "safety", "name": "DynaCertBench System 104 · DCB-OURS-P4-003 · New-8D Parametric Nonlinear Safety", "dynamics": "dx1/dt = x2, dx2/dt = -x1-(1/10)*sin(omega*x1)-x2+x3, dx3/dt = -2*x3+x4, dx4/dt = -(3/2)*x4+x5, dx5/dt = -2*x5+x6, dx6/dt = -(3/2)*x6+(1/5)*x7, dx7/dt = -2*x7+(1/5)*x6+(1/5)*x8-(1/10)*x7*sin(x8), dx8/dt = -(3/2)*x8+(1/5)*x7-(1/10)*x8*cos(x7)", "source": "paper System 104; canonical record DCB-OURS-P4-003", "assumptions": [ "Transcribed from active LaTeX System 104.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "analysis_bounds": [ [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ] ], "fixed_parameters": [ { "name": "omega", "bounds": [ "1/2", 2 ] } ], "initial_set": { "type": "ball", "center": [ 0, 0, 0, 0, 0, 0, 0, 0 ], "radius": "1/2" }, "unsafe_set": { "type": "ball", "center": [ 0, "4/5", 0, 0, 0, 0, 0, 0 ], "radius": "1/10" } }, "candidate": null }, { "record_id": "DCB-OURS-P4-004", "paper_system_number": 105, "paper_system_label": "System 105", "local_name": "S105__DCB-OURS-P4-004", "problem": { "problem_id": "dcb_ours_p4_004", "task_type": "safety", "name": "DynaCertBench System 105 · DCB-OURS-P4-004 · New-5D Arctangent Perturbation Safety", "dynamics": "dx1/dt = x2+(1/2)*x5, dx2/dt = -x1-omega*x2+2*x4, dx3/dt = x4, dx4/dt = -2*x3-(1/8)*atan(x3)-(3/2)*x4-2*x2+x5, dx5/dt = -(1/2)*x1-x4-x5-(1/8)*atan(x5)", "source": "paper System 105; canonical record DCB-OURS-P4-004", "assumptions": [ "Transcribed from active LaTeX System 105.", "Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean." ], "analysis_bounds": [ [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ], [ -1, 1 ] ], "fixed_parameters": [ { "name": "omega", "bounds": [ 1, 2 ] } ], "initial_set": { "type": "ball", "center": [ 0, 0, 0, 0, 0 ], "radius": "1/2" }, "unsafe_set": { "type": "ball", "center": [ 0, 0, "4/5", 0, 0 ], "radius": "1/10" } }, "candidate": null } ], "revision_notes": [ "Generated from active LaTeX systeminstance order.", "All candidates are null; proof answers are not part of the dataset." ] }