{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261116-r02-intermediate-value-theorem-0040",
  "pilot_number": "R0051-26365",
  "number": 26365,
  "family_id": "qf-mod47-ivt-exact-root",
  "title": {
    "zh-Hant": "第 26365 題：介值定理",
    "en": "Question 26365: An Exact Closed-Interval Root Witness Does Not Assert Uniqueness"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261116-r02-intermediate-value-theorem-0040",
    "batch_id": "batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-intermediate-value-theorem"
    ],
    "title_zh": "介值定理",
    "stem_zh": "a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[-2,1],\"a\":1,\"b\":3,\"target\":0,\"root\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(2, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)",
      "explanation_zh": "Wave6第十層核對=(2, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "介值定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod47-ivt-exact-root",
      "seed": 202611162040,
      "parameters": {
        "coefficients": [
          -2,
          1
        ],
        "a": 1,
        "b": 3,
        "target": 0,
        "root": 2,
        "domain": "Exact_polynomial_intermediate_value",
        "operation": "ivt-exact-root",
        "semantic_template": "qf-mod47-ivt-exact-root/{\"a\":[1,1],\"b\":[3,1],\"coefficients\":[[-2,1],[1,1]],\"root\":[2,1],\"target\":[0,1]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261116-r02-intermediate-value-theorem-0040",
    "batch_id": "batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "a6f36fdf27c1402f955dae38cceffdc761d8544659da2e2f7ae46373db14a3f5",
      "family_id": "qf-mod47-ivt-exact-root",
      "seed": 202611162040,
      "semantic_template": "qf-mod47-ivt-exact-root/{\"a\":[1,1],\"b\":[3,1],\"coefficients\":[[-2,1],[1,1]],\"root\":[2,1],\"target\":[0,1]}"
    },
    "fields": {
      "title": "An Exact Closed-Interval Root Witness Does Not Assert Uniqueness",
      "stem": "Coefficients=[-2, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=1<b=3 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=2. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(2, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)",
        "explanation": "Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (2, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)."
      },
      "skills": [
        "An Exact Closed-Interval Root Witness Does Not Assert Uniqueness"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[-2,1],\"a\":1,\"b\":3,\"target\":0,\"root\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Coefficients=[-2, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=1<b=3 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=2. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [ivt-exact-root]",
      "after": "derived = [[root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本頁直接列完整來源欄序，不需JavaScript；contract的6/4/6/6與IVT的8/7/8由原mode選，不補欄也不由答案形狀猜。",
        "en": "List the full source field order without JavaScript.Original modes select contract6/4/6/6 and IVT8/7/8 roles,without padding or answer-shape inference."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "(1)^0",
      "after": "IVT ORIGINAL left endpoint ORIGINAL argument power0 = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "integer_power",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "IVT ORIGINAL left endpoint ORIGINAL argument power0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "IVT ORIGINAL left endpoint coefficient term0 := mul(-2, 1)",
      "after": "IVT ORIGINAL left endpoint coefficient term0 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 IVT ORIGINAL left endpoint coefficient term0，不以小數近似替代。",
          "en": "Recompute IVT ORIGINAL left endpoint coefficient term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-2",
        "label": "IVT ORIGINAL left endpoint coefficient term0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "IVT ORIGINAL left endpoint ascending partial sum0 := sum(0, -2)",
      "after": "IVT ORIGINAL left endpoint ascending partial sum0 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 IVT ORIGINAL left endpoint ascending partial sum0，不以小數近似替代。",
          "en": "Recompute IVT ORIGINAL left endpoint ascending partial sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-2"
        ],
        "output": "-2",
        "label": "IVT ORIGINAL left endpoint ascending partial sum0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "(1)^1",
      "after": "IVT ORIGINAL left endpoint ORIGINAL argument power1 = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "integer_power",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "IVT ORIGINAL left endpoint ORIGINAL argument power1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "IVT ORIGINAL left endpoint coefficient term1 := mul(1, 1)",
      "after": "IVT ORIGINAL left endpoint coefficient term1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 IVT ORIGINAL left endpoint coefficient term1，不以小數近似替代。",
          "en": "Recompute IVT ORIGINAL left endpoint coefficient term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "IVT ORIGINAL left endpoint coefficient term1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "IVT ORIGINAL left endpoint ascending partial sum1 := sum(-2, 1)",
      "after": "IVT ORIGINAL left endpoint ascending partial sum1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 IVT ORIGINAL left endpoint ascending partial sum1，不以小數近似替代。",
          "en": "Recompute IVT ORIGINAL left endpoint ascending partial sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-1",
        "label": "IVT ORIGINAL left endpoint ascending partial sum1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "(3)^0",
      "after": "IVT ORIGINAL right endpoint ORIGINAL argument power0 = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "integer_power",
        "inputs": [
          "3",
          "0"
        ],
        "output": "1",
        "label": "IVT ORIGINAL right endpoint ORIGINAL argument power0"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "IVT ORIGINAL right endpoint coefficient term0 := mul(-2, 1)",
      "after": "IVT ORIGINAL right endpoint coefficient term0 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 IVT ORIGINAL right endpoint coefficient term0，不以小數近似替代。",
          "en": "Recompute IVT ORIGINAL right endpoint coefficient term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-2",
        "label": "IVT ORIGINAL right endpoint coefficient term0"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "IVT ORIGINAL right endpoint ascending partial sum0 := sum(0, -2)",
      "after": "IVT ORIGINAL right endpoint ascending partial sum0 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 IVT ORIGINAL right endpoint ascending partial sum0，不以小數近似替代。",
          "en": "Recompute IVT ORIGINAL right endpoint ascending partial sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-2"
        ],
        "output": "-2",
        "label": "IVT ORIGINAL right endpoint ascending partial sum0"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "(3)^1",
      "after": "IVT ORIGINAL right endpoint ORIGINAL argument power1 = 3",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "integer_power",
        "inputs": [
          "3",
          "1"
        ],
        "output": "3",
        "label": "IVT ORIGINAL right endpoint ORIGINAL argument power1"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "IVT ORIGINAL right endpoint coefficient term1 := mul(1, 3)",
      "after": "IVT ORIGINAL right endpoint coefficient term1 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 IVT ORIGINAL right endpoint coefficient term1，不以小數近似替代。",
          "en": "Recompute IVT ORIGINAL right endpoint coefficient term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "3"
        ],
        "output": "3",
        "label": "IVT ORIGINAL right endpoint coefficient term1"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "IVT ORIGINAL right endpoint ascending partial sum1 := sum(-2, 3)",
      "after": "IVT ORIGINAL right endpoint ascending partial sum1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 IVT ORIGINAL right endpoint ascending partial sum1，不以小數近似替代。",
          "en": "Recompute IVT ORIGINAL right endpoint ascending partial sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "3"
        ],
        "output": "1",
        "label": "IVT ORIGINAL right endpoint ascending partial sum1"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "source roles = [[-2, 1], 1, 3, 0]",
      "after": "derived = [-1, 1, 是, 是]",
      "relation": "implies",
      "rule_id": "IVT_WHOLE_polynomial_continuity_inclusive_both_endpoint_orientations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限多項式的每個monomial與有限和在全interval連續，不從samples猜continuity。target須夾在端點值間，可遞增或遞減，等號亦包含；qualification可報bracketFalse。",
        "en": "Every monomial and finite polynomial sum is continuous on the whole interval,not by sampled agreement.Target bracketing allows increasing or decreasing endpoint values and equality;qualification may report bracketFalse."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "(2)^0",
      "after": "IVT ORIGINAL supplied root ORIGINAL argument power0 = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "integer_power",
        "inputs": [
          "2",
          "0"
        ],
        "output": "1",
        "label": "IVT ORIGINAL supplied root ORIGINAL argument power0"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "IVT ORIGINAL supplied root coefficient term0 := mul(-2, 1)",
      "after": "IVT ORIGINAL supplied root coefficient term0 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 IVT ORIGINAL supplied root coefficient term0，不以小數近似替代。",
          "en": "Recompute IVT ORIGINAL supplied root coefficient term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-2",
        "label": "IVT ORIGINAL supplied root coefficient term0"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "IVT ORIGINAL supplied root ascending partial sum0 := sum(0, -2)",
      "after": "IVT ORIGINAL supplied root ascending partial sum0 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 IVT ORIGINAL supplied root ascending partial sum0，不以小數近似替代。",
          "en": "Recompute IVT ORIGINAL supplied root ascending partial sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-2"
        ],
        "output": "-2",
        "label": "IVT ORIGINAL supplied root ascending partial sum0"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "(2)^1",
      "after": "IVT ORIGINAL supplied root ORIGINAL argument power1 = 2",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "integer_power",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "IVT ORIGINAL supplied root ORIGINAL argument power1"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "IVT ORIGINAL supplied root coefficient term1 := mul(1, 2)",
      "after": "IVT ORIGINAL supplied root coefficient term1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 IVT ORIGINAL supplied root coefficient term1，不以小數近似替代。",
          "en": "Recompute IVT ORIGINAL supplied root coefficient term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "IVT ORIGINAL supplied root coefficient term1"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "IVT ORIGINAL supplied root ascending partial sum1 := sum(-2, 2)",
      "after": "IVT ORIGINAL supplied root ascending partial sum1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 IVT ORIGINAL supplied root ascending partial sum1，不以小數近似替代。",
          "en": "Recompute IVT ORIGINAL supplied root ascending partial sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "2"
        ],
        "output": "0",
        "label": "IVT ORIGINAL supplied root ascending partial sum1"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "source roles = [1, 3, 0, 2]",
      "after": "derived = [0, 是, 是, 否, 否]",
      "relation": "implies",
      "rule_id": "IVT_actual_closed_interval_root_hit_endpoint_flag_separate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐供給root核f(root)=target與閉intervalmembership。endpoint family也可給interior root，此flag可False；exact-root的固定No只不保證一般unique／rational closed form，不否定此witness。",
        "en": "Check the supplied exact root value and closed-interval membership.An endpoint-family witness may be interior,with flagFalse.The fixed No on exact-root withholds general uniqueness/rational closed-form guarantees,not this witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "2",
      "0",
      true,
      true,
      false,
      "IVT_existence_does_not_imply_uniqueness_or_rational_closed_form"
    ]
  },
  "conclusion": {
    "canonical_value": "(2, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations/questions.jsonl",
      "batch_sha256": "3f5cf45e49b04c091326c1359f302c28b5c81c4cfe741d9b65ddca27c2094fd6",
      "original_line": 40,
      "original_line_sha256": "9e820ea922c65a0b0883ca67d58417367bbe61dcbeb652d9c9a7207852a9667f",
      "question_compact_sha256": "a6f36fdf27c1402f955dae38cceffdc761d8544659da2e2f7ae46373db14a3f5",
      "english_row_compact_sha256": "0785740c4668367b4c4f274019ef0d82750b45629f9d5e7015f7adf448c9bac0",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-intermediate-value-theorem": {
        "sha256": "52c0e9f2fe41f93f083f280990d519515da4d4698c3a483bd5f1a57ff44a71e7",
        "english_sha256": "cee1f784671e6c0f5cbea99c3cd16177aa90173e1b2f55fa7ac4f5117cfa7f94",
        "original_path": "public/data/mko/intermediate-value-theorem.json",
        "english_path": "public/data/i18n/editions/mko-intermediate-value-theorem.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "74f6f34e0a695e06f9d836b5f5c677f9f716d2291c75c8f2f5b809ff0935bcff",
    "original_candidate_sha256": "5c59ec4aeb7f8d06a11bf1df01093033cfba7d4715cb4d784fe6d246ba23207f",
    "original_candidate_line_sha256": "5c59ec4aeb7f8d06a11bf1df01093033cfba7d4715cb4d784fe6d246ba23207f"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-010-p52",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3",
      "question_first_witness_sha256": "ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 21,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "7b9362a43d60eee13ec69cf51a8845fde14ce6a7ccd42e9adcbeacf69e311e80",
      "arithmetic_steps": 18
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}