{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-intermediate-value-theorem-01",
  "mko_id": "mko-intermediate-value-theorem",
  "title": {
    "zh-Hant": "多項式 bracket 與精確中間值",
    "en": "A polynomial bracket and an exact intermediate value"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-intermediate-value-theorem-01",
    "source": {
      "mko_id": "mko-intermediate-value-theorem",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "52c0e9f2fe41f93f083f280990d519515da4d4698c3a483bd5f1a57ff44a71e7",
      "english_sha256": "cee1f784671e6c0f5cbea99c3cd16177aa90173e1b2f55fa7ac4f5117cfa7f94"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "algebra",
      "polynomial",
      "IVT"
    ],
    "goal": {
      "zh-Hant": "對 f(x)=x^2-1 在 [0,2] 找值 0，並保留端點及非唯一邊界。",
      "en": "Find value zero for f(x)=x^2-1 on [0,2], retaining endpoint and nonuniqueness boundaries."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "f(x)=x^2-1，a=0,b=2，目標 y=0。",
          "en": "f(x)=x^2-1, a=0,b=2, target y=0."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "a=0,\\ b=2",
        "after_tex": "a<b,\\quad[a,b]=[0,2]\\text{ is the full domain interval}",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先固定完整閉區間，不把零散點集當區間。",
          "en": "Fix the whole closed interval rather than substituting a discrete point set."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "a=b 不在此應用前提。",
            "en": "The degenerate a=b case is outside this application."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "f(x)=x^2-1",
        "after_tex": "f\\text{ continuous at every point of }[0,2]",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整多項式保證全閉區間連續，並非樣本看似平滑。",
          "en": "The complete polynomial guarantees continuity on the whole closed interval, not apparent smoothness of samples."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "連續性的全稱範圍明確。",
            "en": "The universal range of continuity is explicit."
          }
        },
        "leaf_basis": "polynomial"
      },
      {
        "id": "S03",
        "uses": [
          "G1"
        ],
        "before_tex": "f(0)=0^2-1,\\quad f(2)=2^2-1",
        "after_tex": "f(0)=-1\\le0\\le3=f(2)",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "精確核目標在端點值之間。",
          "en": "Exactly verify the target lies between the endpoint values."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "bracket 是包含端點的非嚴格條件。",
            "en": "Bracketing is inclusive, allowing endpoint equality."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S04",
        "uses": [
          "S01",
          "S02",
          "S03"
        ],
        "before_tex": "f\\text{ continuous on }[0,2],\\quad-1\\le0\\le3",
        "after_tex": "\\exists c\\in[0,2]:f(c)=0",
        "relation": "theorem_application",
        "rule": "IVT",
        "conditions": [],
        "explanation": {
          "zh-Hant": "明示套用 IVT 前提只得到存在，不包含唯一或可寫有理式。",
          "en": "The explicitly qualified IVT gives existence only, not uniqueness or a rational formula."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "每個前提都由前步給出，不由端點變號單獨跳結論。",
            "en": "Every premise is supplied; endpoint sign change alone does not give the conclusion."
          }
        },
        "leaf_basis": "IVT"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04"
        ],
        "before_tex": "c=1",
        "after_tex": "0<1<2,\\quad f(1)=1^2-1=0",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "此多項式另有精確有理 witness，獨立核範圍與等式。",
          "en": "This polynomial additionally has an exact rational witness, with interval membership and equality checked independently."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "個別有理根不代表所有 IVT 問題都有有理閉式。",
            "en": "One rational root does not give rational formulas for every IVT problem."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S06",
        "uses": [
          "G1"
        ],
        "before_tex": "[a,b]=[1,2],\\quad y=0",
        "after_tex": "c=a=1\\text{ is a valid endpoint witness}",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "IVT 的 c 可在端點，這與 MVT 的嚴格內點責任不同。",
          "en": "IVT permits c at an endpoint, unlike the strict-interior MVT obligation."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "另分支仍 a<b 且 f(1)=y。",
            "en": "The separate branch still satisfies a<b and f(1)=y."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S07",
        "uses": [
          "G1"
        ],
        "before_tex": "g(x)=x^3-x\\text{ on }[-2,2]",
        "after_tex": "g(-2)=-6,\\ g(2)=6,\\quad g(-1)=g(0)=g(1)=0",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [],
        "explanation": {
          "zh-Hant": "明示另一連續 bracket 例具有三個不同根，保護非唯一界線。",
          "en": "Another continuous bracketed example has three distinct roots, protecting the nonuniqueness boundary."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "不是宣稱原 f 在 [0,2] 有三根；例子分清。",
            "en": "This does not assert three roots for the original f on [0,2]; the examples are distinct."
          }
        },
        "leaf_basis": "polynomial"
      },
      {
        "id": "S08",
        "uses": [
          "G1"
        ],
        "before_tex": "s(x)=\\begin{cases}-1&x<0\\\\1&x\\ge0\\end{cases}\\quad\\text{on }[-1,1]",
        "after_tex": "s(-1)<0<s(1),\\quad0\\notin s([-1,1])",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "不連續跳躍可變號卻不取零；連續性前提不能刪除。",
          "en": "A discontinuous jump changes sign without taking zero, so continuity cannot be omitted."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "失敗在 0 的連續性，端點 bracket 本身確實成立。",
            "en": "Continuity fails at zero, while the endpoint bracket itself genuinely holds."
          }
        },
        "leaf_basis": "algebra"
      }
    ],
    "result": {
      "zh-Hant": "主例 witness c=1；IVT 只保證存在，可在端點且一般不唯一。",
      "en": "The main witness is c=1; IVT guarantees existence only, permits endpoints and need not give uniqueness."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "端點變號即使不連續仍必有根。",
          "en": "Sign change gives a root even without continuity."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "S08 跳躍函數不取零。",
          "en": "The jump at S08 never takes zero."
        }
      },
      {
        "claim": {
          "zh-Hant": "IVT 的 witness 必嚴格在內部。",
          "en": "An IVT witness must lie strictly inside."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "S06 的端點值已等於目標。",
          "en": "The endpoint at S06 already equals the target."
        }
      },
      {
        "claim": {
          "zh-Hant": "存在就必唯一。",
          "en": "Existence guarantees uniqueness."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "S07 的三個根反駁。",
          "en": "The three roots at S07 refute it."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "固定a<b、完整interval domain與continuous polynomial certificate，target y須inclusive介於f(a)、f(b)。exact families提供明示c∈[a,b]且f(c)=y；endpoint root保留。IVT只保證存在，不保證唯一、rational closed form或有限bisection已收斂。明示discontinuous step可有sign change但沒有root；有限samples不證continuity。",
          "en": "Fix a<b, the entire interval domain and a continuous polynomial certificate. The target y must lie inclusively between f(a) and f(b), in either orientation. Exact families supply c∈[a,b] with f(c)=y and retain endpoint roots. IVT gives existence, not uniqueness, a rational closed form or convergence of finitely many bisection steps. An explicit discontinuous step can change sign without taking the target value, and finite samples do not prove continuity."
        },
        "source": {
          "mko_id": "mko-intermediate-value-theorem",
          "sha256": "52c0e9f2fe41f93f083f280990d519515da4d4698c3a483bd5f1a57ff44a71e7",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "algebra",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。",
          "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities."
        }
      },
      {
        "id": "polynomial",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "多項式在實數上連續且可逐項微分；d(x^k)/dx=k x^(k-1)，常數導數為零。",
          "en": "Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero."
        }
      },
      {
        "id": "IVT",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "若 a<b、f 在整個 [a,b] 連續且 y 介於兩端值（含相等、任一方向），則存在 c∈[a,b] 使 f(c)=y。",
          "en": "If a<b and f is continuous throughout [a,b], every y between its endpoint values inclusively and in either orientation equals f(c) for some c in [a,b]."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "IVT-witness",
        "lhs": "1**2-1",
        "rhs": "0",
        "step": "S05"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整多項式證書與精確 witness；不把有限 bisection 稱作極限證明。",
      "en": "A full polynomial certificate and exact witness, not a finite-bisection convergence claim."
    },
    "md_symbols": [
      [
        "y",
        "目標值",
        "target value"
      ],
      [
        "c",
        "包含端點的 witness",
        "witness allowing endpoints"
      ]
    ],
    "md_error_controls": [
      {
        "title": "前提逐項 / Separate premises",
        "zh": "完整區間、連續性及 bracket 各自核。",
        "en": "Check the full interval, continuity and bracket separately."
      },
      {
        "title": "有限演算法 / Finite algorithms",
        "zh": "有限二分只給區間或近似，不自動是無限收斂證明。",
        "en": "Finite bisection gives a bracket or approximation, not automatically an infinite convergence proof."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "374944ef5526713f1c77728586ee43f3af567de157b811fc6e0370d3635c1ff7",
    "original_source_sha256": "374944ef5526713f1c77728586ee43f3af567de157b811fc6e0370d3635c1ff7",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}