{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261008-r04-derivative-0099",
  "pilot_number": "R0016-08599",
  "number": 8599,
  "family_id": "qf-derivative-shifted-absolute",
  "title": {
    "zh-Hant": "第 8599 題：平移絕對值的導數與角點",
    "en": "Question 8599: Derivative and Corner of a Shifted Absolute Value"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261008-r04-derivative-0099",
    "batch_id": "batch-20261007-advance-20261008-r04-derivative-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-derivative"
    ],
    "title_zh": "平移絕對值的導數與角點",
    "stem_zh": "全實數論域，以下函數／點均以明示實數模型解讀。f(x)=|x−(11)|，求 f′(-12)；若不存在則回答「不存在」。",
    "answer": {
      "kind": "short_text",
      "value": "-1",
      "explanation_zh": "相對位置=-23；導數=-1。離角點按左右位置取斜率，在角點兩側導數不相同；不能把對稱有限差商零當導數。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "平移絕對值的導數與角點"
      ]
    },
    "variant": {
      "family_id": "qf-derivative-shifted-absolute",
      "seed": 202610084099,
      "parameters": {
        "c": 11,
        "x": -12,
        "operation": "shifted_absolute_derivative",
        "domain": "reals",
        "semantic_template": "qf-derivative-shifted-absolute/{\"inputs\":[\"-23\"],\"operation\":\"shifted_absolute_derivative\"}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-07",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261007-advance-20261008-r04-derivative-0099",
    "batch_id": "batch-20261007-advance-20261008-r04-derivative-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "9222e4df303274618b1ea4d15b3b81df87231c60a83da2516dd02756e2b88b49",
      "family_id": "qf-derivative-shifted-absolute",
      "seed": 202610084099,
      "semantic_template": "qf-derivative-shifted-absolute/{\"inputs\":[\"-23\"],\"operation\":\"shifted_absolute_derivative\"}"
    },
    "fields": {
      "title": "Derivative and Corner of a Shifted Absolute Value",
      "stem": "On all R, f(t)=|t−(11)|. Find its derivative at x=-12, or answer does not exist at the corner.",
      "answer": {
        "kind": "short_text",
        "value": "-1",
        "explanation": "Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative."
      },
      "skills": [
        "Derivative and Corner of a Shifted Absolute Value"
      ],
      "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": "全實數論域，以下函數／點均以明示實數模型解讀。f(x)=|x−(11)|，求 f′(-12)；若不存在則回答「不存在」。",
        "en": "On all R, f(t)=|t−(11)|. Find its derivative at x=-12, or answer does not exist at the corner."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared real formula on all R;derivative means the two-sided h->0 quotient limit",
      "after": "finite-step evaluation and derivative existence are separate",
      "relation": "implies",
      "rule_id": "derivative_limit_definition_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "導數由[f(x+h)−f(x)]/h在h→0、h≠0的雙側極限定義；有限一次代值不是任意函數可微的證據。",
        "en": "A derivative is the two-sided limit of[f(x+h)−f(x)]/h as h→0 with h≠0.One finite evaluation is not evidence that an arbitrary function is differentiable."
      },
      "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": "relative position x-c := sub(-12, 11)",
      "after": "relative position x-c = -23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步的原始操作數與運算順序精確計算；除法前須先核分母非零。",
        "en": "Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 relative position x-c，不以小數近似替代。",
          "en": "Recompute relative position x-c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-12",
          "11"
        ],
        "output": "-23",
        "label": "relative position x-c"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "sign-preserving neighborhood radius := div(23, 2)",
      "after": "sign-preserving neighborhood radius = 23/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步的原始操作數與運算順序精確計算；除法前須先核分母非零。",
        "en": "Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sign-preserving neighborhood radius，不以小數近似替代。",
          "en": "Recompute sign-preserving neighborhood radius with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "2"
        ],
        "output": "23/2",
        "label": "sign-preserving neighborhood radius"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "x-c=-23;0<|h|<23/2",
      "after": "x+h-c keeps sign;quotient=-1",
      "relation": "implies",
      "rule_id": "absolute_off_corner_local_sign",
      "conditions": [],
      "explanation": {
        "zh-Hant": "取半個到角點距離作鄰域半徑，x+h−c與x−c同號，所以絕對值在此鄰域是同一個仿射分支，差商固定為相應斜率。",
        "en": "Choose half the distance to the corner as the radius.Then x+h−c has the same sign as x−c,and the absolute value uses one affine branch throughout that neighborhood,with constant corresponding quotient."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "local constant quotient=-1;radius=23/2",
      "after": "two-sided derivative at -12=-1",
      "relation": "implies",
      "rule_id": "absolute_off_corner_limit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "局部差商恆定給出雙側極限；結論不套到被排除的角點。",
        "en": "The locally constant quotient gives the two-sided limit.This conclusion does not apply at the excluded corner."
      },
      "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": -1
  },
  "conclusion": {
    "canonical_value": "-1",
    "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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      "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261008-r04-derivative-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261008-r04-derivative-foundations.en.jsonl"
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        "original_path": "public/data/mko/derivative.json",
        "english_path": "public/data/i18n/editions/mko-derivative.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "8e8954cf690c44da1ec1efa08994a485c73ca8706b7865d854e3f2cdfd4ca89f",
    "original_candidate_sha256": "e30fe2c18930da1d227b436d9d7cb180ae603644168164c25bf23cab29dc68fb",
    "original_candidate_line_sha256": "e30fe2c18930da1d227b436d9d7cb180ae603644168164c25bf23cab29dc68fb"
  },
  "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-20261026-005-p17",
    "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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    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 5,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
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    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}