{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261110-r02-mean-value-theorem-0085",
  "pilot_number": "R0043-22210",
  "number": 22210,
  "family_id": "qf-mod41-mvt-failure-boundary",
  "title": {
    "zh-Hant": "第 22210 題：微分中值定理",
    "en": "Question 22210: Failed MVT Hypotheses Need Their Own Exact Output Layout"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261110-r02-mean-value-theorem-0085",
    "batch_id": "batch-20261014-advance-20261110-r02-mean-value-theorem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-mean-value-theorem"
    ],
    "title_zh": "微分中值定理",
    "stem_zh": "f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"degenerate_interval\",\"a\":9,\"b\":9,\"scale\":10,\"offset\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(degenerate_interval, 9, 9, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)",
      "explanation_zh": "Wave6第四層核對=(degenerate_interval, 9, 9, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "微分中值定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod41-mvt-failure-boundary",
      "seed": 202611102085,
      "parameters": {
        "mode": "degenerate_interval",
        "a": 9,
        "b": 9,
        "scale": 10,
        "offset": -9,
        "domain": "Exact_polynomial_mean_value_theorem",
        "operation": "mvt-failure-boundary",
        "semantic_template": "qf-mod41-mvt-failure-boundary/{\"a\":9,\"b\":9,\"mode\":\"degenerate_interval\",\"offset\":-9,\"scale\":10}"
      }
    },
    "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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261110-r02-mean-value-theorem-0085",
    "batch_id": "batch-20261014-advance-20261110-r02-mean-value-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "521642a62fde12c9234c009aec5fa1a1c423b97486b59b69a066a29a41146c9b",
      "family_id": "qf-mod41-mvt-failure-boundary",
      "seed": 202611102085,
      "semantic_template": "qf-mod41-mvt-failure-boundary/{\"a\":9,\"b\":9,\"mode\":\"degenerate_interval\",\"offset\":-9,\"scale\":10}"
    },
    "fields": {
      "title": "Failed MVT Hypotheses Need Their Own Exact Output Layout",
      "stem": "Use boundary mode=degenerate_interval, a=9, b=9, scale=10, offset=-9. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, ordered_endpoints, secant_slope_or_null, theorem_applicability, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(degenerate_interval, 9, 9, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required)",
        "explanation": "Use the original mode-selected fields, without padding. Degenerate/reversed endpoint mode withholds a secant value as null; even if a>b makes a numerical quotient possible, it fails the ordered-interval theorem scope. A jump fails continuity and differentiability at zero; a cusp is continuous but not differentiable there. No means this failed-hypothesis case cannot invoke MVT, not a general classification of every unrelated function. The exact result, retaining the requested order, is (degenerate_interval, 9, 9, No, null, No, a_less_than_b_and_nonzero_secant_denominator_required)."
      },
      "skills": [
        "Failed MVT Hypotheses Need Their Own Exact Output Layout"
      ],
      "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": "f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"degenerate_interval\",\"a\":9,\"b\":9,\"scale\":10,\"offset\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use boundary mode=degenerate_interval, a=9, b=9, scale=10, offset=-9. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, ordered_endpoints, secant_slope_or_null, theorem_applicability, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=mvt-failure-boundary;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[mode, a, b, ordered_endpoints, secant_slope_or_null, theorem_applicability, boundary]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題歷史介面標記保留，直接列來源有序欄位，不需JavaScript；MVT由原mode選欄序，不補空欄或看答案形狀猜。tag與固定No保原聲稱範圍。",
        "en": "Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope."
      },
      "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": "ORIGINAL mode=degenerate_interval,a=9,b=9,scale=10,offset=-9",
      "after": "ordered a<b=否;source secant value withheld=null;theorem applicability=否;keep ORIGINAL7-field layout even if reversed endpoints allow an arithmetic quotient",
      "relation": "implies",
      "rule_id": "mvt_degenerate_or_reversed_scope_null_not_quotient_repair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本定理明示a<b；相同端點分母0，反向端點則不合原有序區間慣例，仍依來源保null，不擅自交換端點或補成9欄。這是適用範圍，不是否認其他可重述的定理。",
        "en": "This theorem declares a<b.Equal endpoints give denominator0;reversed endpoints fail its ordered-interval convention.Keep source null without swapping endpoints or padding to9 fields.This is a scope judgment,not a denial of other formulations."
      },
      "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": [
      "degenerate_interval",
      9,
      9,
      false,
      null,
      false,
      "a_less_than_b_and_nonzero_secant_denominator_required"
    ]
  },
  "conclusion": {
    "canonical_value": "(degenerate_interval, 9, 9, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)",
    "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": {
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    "question_binding": {
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      "batch_sha256": "8ba45e49fc1080aa7ff54d64afad99686fb70f44ad68b49ff9bf82c95b3f3256",
      "original_line": 85,
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      "question_compact_sha256": "521642a62fde12c9234c009aec5fa1a1c423b97486b59b69a066a29a41146c9b",
      "english_row_compact_sha256": "b7981e617a2b9806ab0ab992bd1673b106f5aa5ef0d1eb0194624303bf1c62eb",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261110-r02-mean-value-theorem-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261110-r02-mean-value-theorem-foundations.en.jsonl"
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        "original_path": "public/data/mko/mean-value-theorem.json",
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    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "747b98e75085a6b55251b2d7e04bbf7c60cb5500e5047439b2798fb5f45538fa",
    "original_candidate_sha256": "46b365d3ea732d25a7ee6e53170238f7ceaa8abe9f080b8b02190efcad9d14b3",
    "original_candidate_line_sha256": "46b365d3ea732d25a7ee6e53170238f7ceaa8abe9f080b8b02190efcad9d14b3"
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      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
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      "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-20261026-009-p44",
    "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": {
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}