{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261116-r02-intermediate-value-theorem-0083",
  "pilot_number": "R0051-26308",
  "number": 26308,
  "family_id": "qf-mod47-ivt-discontinuity-boundary",
  "title": {
    "zh-Hant": "第 26308 題：介值定理",
    "en": "Question 26308: A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261116-r02-intermediate-value-theorem-0083",
    "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-discontinuity-boundary；資料={\"mode\":\"nonbracket\",\"scale\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(nonbracket, f(x)=8*x, 0, 1, 16, 否, 否)",
      "explanation_zh": "Wave6第十層核對=(nonbracket, f(x)=8*x, 0, 1, 16, 否, 否)；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": 1,
      "skills": [
        "介值定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod47-ivt-discontinuity-boundary",
      "seed": 202611162083,
      "parameters": {
        "mode": "nonbracket",
        "scale": 8,
        "domain": "Exact_polynomial_intermediate_value",
        "operation": "ivt-discontinuity-boundary",
        "semantic_template": "qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonbracket\",\"scale\":[8,1]}"
      }
    },
    "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-0083",
    "batch_id": "batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4a94b70d67e65df3bbc84c5cb08e9412349a54954691a7764293e1d0e074ced3",
      "family_id": "qf-mod47-ivt-discontinuity-boundary",
      "seed": 202611162083,
      "semantic_template": "qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonbracket\",\"scale\":[8,1]}"
    },
    "fields": {
      "title": "A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries",
      "stem": "Use boundary mode=nonbracket and positive exact scale=8. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).",
      "answer": {
        "kind": "expression",
        "value": "(nonbracket, f(x)=8*x, 0, 1, 16, No, No)",
        "explanation": "The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=8*x, 0, 1, 16, No, No)."
      },
      "skills": [
        "A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries"
      ],
      "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-discontinuity-boundary；資料={\"mode\":\"nonbracket\",\"scale\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use boundary mode=nonbracket and positive exact scale=8. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [ivt-discontinuity-boundary]",
      "after": "derived = [[mode, function_description, a, b, target, inclusive_bracket, root_exists]]",
      "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": "IVT nonbracket ORIGINAL twice scale := mul(2, 8)",
      "after": "IVT nonbracket ORIGINAL twice scale = 16",
      "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 nonbracket ORIGINAL twice scale，不以小數近似替代。",
          "en": "Recompute IVT nonbracket ORIGINAL twice scale with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "8"
        ],
        "output": "16",
        "label": "IVT nonbracket ORIGINAL twice scale"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [nonbracket, 8]",
      "after": "derived = [nonbracket, f(x)=8*x, 0, 1, 16, 否, 否]",
      "relation": "implies",
      "rule_id": "IVT_original_jump_nonbracket_nonunique_distinct_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "jump只取±s所以不中0，雖端點bracket卻不continuous；scaled identity在[0,1]值只到s而target2s不bracket；x²在[-s,s]對s²有兩endpoint roots，存在不保唯一。各False欄意義依mode，不補continuous欄。",
        "en": "The jump takes only±s and misses0 despite endpoint bracketing because continuity fails.The scaled identity on[0,1] reaches at most s,not2s.For x² on[-s,s],target s² has two endpoint roots,so existence is not uniqueness.False roles follow the mode;do not pad a continuity field."
      },
      "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": [
      "nonbracket",
      "f(x)=8*x",
      0,
      1,
      "16",
      false,
      false
    ]
  },
  "conclusion": {
    "canonical_value": "(nonbracket, f(x)=8*x, 0, 1, 16, 否, 否)",
    "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": 83,
      "original_line_sha256": "de97d488a4be1774b22a0e8fd5b2aa8f4bcf339ae8c6f0618b71e4dba19e89c6",
      "question_compact_sha256": "4a94b70d67e65df3bbc84c5cb08e9412349a54954691a7764293e1d0e074ced3",
      "english_row_compact_sha256": "b736df991d37082ab8bf96770e1acd2d726263694a714fa4ecdc2008c0e20fce",
      "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": "c3e5527f3d37817c4e5fd712a383778ddf859dcc295fb595096a90486ee5c491",
    "original_candidate_line_sha256": "c3e5527f3d37817c4e5fd712a383778ddf859dcc295fb595096a90486ee5c491"
  },
  "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": 3,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "39171e7fcf2f9be8efd25f69ff314510736fc856d0d502689387a27e7c2ae8b3",
      "arithmetic_steps": 1
    },
    "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."
  }
}