{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261021-r05-bisection-method-0024",
  "pilot_number": "R0003-02024",
  "number": 2024,
  "family_id": "qf-app21-bisection-bracket-qualification",
  "title": {
    "zh-Hant": "第 2024 題：有限二分bracket",
    "en": "Question 2024: Qualify Declared Sign Data without an Existence Claim"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261021-r05-bisection-method-0024",
    "batch_id": "batch-20261013-advance-20261021-r05-bisection-method-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-bisection-method"
    ],
    "title_zh": "有限二分bracket",
    "stem_zh": "已提供的符號 bracket 資料。操作=bisection-bracket-qualification；資料={\"left\":[23,1],\"right\":[27,1],\"left_sign\":-1,\"right_sign\":1}。請依 [left,right,left_sign,right_sign,declared_sign_bracket] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(23, 27, -1, 1, 是)",
      "explanation_zh": "應用擴展核對=(23, 27, -1, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不由異號自行推出根存在；只核對有限更新與寬度。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限二分bracket"
      ]
    },
    "variant": {
      "family_id": "qf-app21-bisection-bracket-qualification",
      "seed": 202610215024,
      "parameters": {
        "left": [
          23,
          1
        ],
        "right": [
          27,
          1
        ],
        "left_sign": -1,
        "right_sign": 1,
        "domain": "Declared_sign_bracket_updates",
        "operation": "bisection-bracket-qualification",
        "semantic_template": "qf-app21-bisection-bracket-qualification/{\"left\":[23,1],\"left_sign\":-1,\"right\":[27,1],\"right_sign\":1}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261021-r05-bisection-method-0024",
    "batch_id": "batch-20261013-advance-20261021-r05-bisection-method-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ea298075cd0feef084f5ef87cb47f96564bce1aaca3c0a682a6a15bc3cc1c5ff",
      "family_id": "qf-app21-bisection-bracket-qualification",
      "seed": 202610215024,
      "semantic_template": "qf-app21-bisection-bracket-qualification/{\"left\":[23,1],\"left_sign\":-1,\"right\":[27,1],\"right_sign\":1}"
    },
    "fields": {
      "title": "Qualify Declared Sign Data without an Existence Claim",
      "stem": "Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=23/1, right=27/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).",
      "answer": {
        "kind": "expression",
        "value": "(23, 27, -1, 1, Yes)",
        "explanation": "Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (23, 27, -1, 1, Yes)."
      },
      "skills": [
        "Qualify Declared Sign Data without an Existence Claim"
      ],
      "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": "已提供的符號 bracket 資料。操作=bisection-bracket-qualification；資料={\"left\":[23,1],\"right\":[27,1],\"left_sign\":-1,\"right_sign\":1}。請依 [left,right,left_sign,right_sign,declared_sign_bracket] 欄序回答。",
        "en": "Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=23/1, right=27/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "left=23<right=27",
      "after": "all endpoints and midpoints are exact rationals",
      "relation": "implies",
      "rule_id": "ordered_bracket_data",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核對端點嚴格有序；這些是已提供的資料，不代表已呼叫某個未知函數。",
        "en": "Check strict endpoint order first;these are supplied data,not observations from an evaluated unknown function."
      },
      "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": "left sign=-1; right sign=1",
      "after": "declared sign-bracket condition=是",
      "relation": "implies",
      "rule_id": "recorded_sign_qualification",
      "conditions": [],
      "explanation": {
        "zh-Hant": "判準是任一端零符號，或兩個非零符號異號；不是由區間長度推斷。",
        "en": "The criterion is a zero endpoint sign or opposite nonzero signs,not interval length."
      },
      "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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "a sign-bracket record is not a verified continuous function",
      "after": "root existence needs the additional continuity/evidence premises",
      "relation": "implies",
      "rule_id": "sign_data_not_IVT_proof",
      "conditions": [],
      "explanation": {
        "zh-Hant": "異號紀錄本身不證根存在；不連續的跳躍函數可異號卻不取0。",
        "en": "An opposite-sign record alone does not establish root existence;a discontinuous jump may change sign without attaining0."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "failure of this sign criterion",
      "after": "does not prove absence of an interior root",
      "relation": "implies",
      "rule_id": "qualification_not_nonexistence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同號端點也可能有內部根，例如x²−1在[−2,2]；本題只分類宣告的符號紀錄。",
        "en": "Same-sign endpoints may still have interior roots,such as x²−1 on[−2,2];this task classifies only the declared signs."
      },
      "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": [
      "23",
      "27",
      -1,
      1,
      true
    ],
    "decision": "qualification"
  },
  "conclusion": {
    "canonical_value": "(23, 27, -1, 1, 是)",
    "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."
  },
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      "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."
    },
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    "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.",
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}