{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261008-r03-continuity-0089",
  "pilot_number": "R0005-03389",
  "number": 3389,
  "family_id": "qf-continuity-point-defects",
  "title": {
    "zh-Hant": "第 3389 題：左右極限相對點值的缺陷",
    "en": "Question 3389: One-Sided Limit Defects Relative to the Point Value"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261008-r03-continuity-0089",
    "batch_id": "batch-20261007-advance-20261008-r03-continuity-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-continuity"
    ],
    "title_zh": "左右極限相對點值的缺陷",
    "stem_zh": "全實數論域，以下函數／點均以明示實數模型解讀。當 x<1 時 f(x)=2x+(12)，x>1 時 f(x)=-2x+(-10)，點值 f(1)=1。求 (左極限−點值, 右極限−點值)。",
    "answer": {
      "kind": "expression",
      "value": "(13, -13)",
      "explanation_zh": "點缺陷=(13, -13)。在明示兩側仿射模型中求精確差，斜率與共同平移不另創造相同缺陷的變種。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "左右極限相對點值的缺陷"
      ]
    },
    "variant": {
      "family_id": "qf-continuity-point-defects",
      "seed": 202610083089,
      "parameters": {
        "a": 1,
        "left_m": 2,
        "left_b": 12,
        "right_m": -2,
        "right_b": -10,
        "value": 1,
        "operation": "point_defects",
        "domain": "reals",
        "semantic_template": "qf-continuity-point-defects/{\"inputs\":[\"13\",\"-13\"],\"operation\":\"point_defects\"}"
      }
    },
    "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-r03-continuity-0089",
    "batch_id": "batch-20261007-advance-20261008-r03-continuity-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "61b9dbfe8b9a1ccf89012d5b3488ccb06e741798951a3f40f385bb70a2d76c3f",
      "family_id": "qf-continuity-point-defects",
      "seed": 202610083089,
      "semantic_template": "qf-continuity-point-defects/{\"inputs\":[\"13\",\"-13\"],\"operation\":\"point_defects\"}"
    },
    "fields": {
      "title": "One-Sided Limit Defects Relative to the Point Value",
      "stem": "On all R, The left branch at x<1 is (2)x+(12); the right branch at x>1 is (-2)x+(-10). The center value is 1. Give (left limit−center value, right limit−center value).",
      "answer": {
        "kind": "expression",
        "value": "(13, -13)",
        "explanation": "The side limits are 14 and -12; subtract the independently given center value to obtain (13, -13). Both defects must be zero for continuity in this model."
      },
      "skills": [
        "One-Sided Limit Defects Relative to the Point 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。當 x<1 時 f(x)=2x+(12)，x>1 時 f(x)=-2x+(-10)，點值 f(1)=1。求 (左極限−點值, 右極限−點值)。",
        "en": "On all R, The left branch at x<1 is (2)x+(12); the right branch at x>1 is (-2)x+(-10). The center value is 1. Give (left limit−center value, right limit−center value)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared real model;center a=1",
      "after": "distinguish punctured limits,point values and a new extension",
      "relation": "implies",
      "rule_id": "continuity_model_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只使用明示實數公式；點值、去心極限及另建延拓的值不是同一欄位。",
        "en": "Use the declared real formulas;point values,punctured limits and values of a newly defined extension are different data."
      },
      "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 branch slope times center := mul(2, 1)",
      "after": "left branch slope times center = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將中心a代入該仿射公式的乘法部分。",
        "en": "Substitute the center a into this affine formula's product term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left branch slope times center，不以小數近似替代。",
          "en": "Recompute left branch slope times center with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "left branch slope times center"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "left branch center value := sum(2, 12)",
      "after": "left branch center value = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加本分支截距，不混用另一側的常數。",
        "en": "Then add this branch's intercept,not the other branch's constant."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left branch center value，不以小數近似替代。",
          "en": "Recompute left branch center value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "12"
        ],
        "output": "14",
        "label": "left branch center value"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "right branch slope times center := mul(-2, 1)",
      "after": "right branch slope times center = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將中心a代入該仿射公式的乘法部分。",
        "en": "Substitute the center a into this affine formula's product term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 right branch slope times center，不以小數近似替代。",
          "en": "Recompute right branch slope times center with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-2",
        "label": "right branch slope times center"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "right branch center value := sum(-2, -10)",
      "after": "right branch center value = -12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加本分支截距，不混用另一側的常數。",
        "en": "Then add this branch's intercept,not the other branch's constant."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 right branch center value，不以小數近似替代。",
          "en": "Recompute right branch center value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "-10"
        ],
        "output": "-12",
        "label": "right branch center value"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "left L=14;right R=-12;given f(a)=1",
      "after": "these are one-sided limits,not automatic point substitutions into f(a)",
      "relation": "implies",
      "rule_id": "affine_one_sided_limits",
      "conditions": [],
      "explanation": {
        "zh-Hant": "仿射分支與其中心候選L的差為m(x−a)；任意ε>0取δ=ε/(1+|m|)，分支在相應一側便趨L。中心點值仍取明示v。",
        "en": "For an affine branch the difference from its center limit L is m(x−a).For any ε>0 choose δ=ε/(1+|m|) to obtain the one-sided limit.The actual center value remains the separately specified v."
      },
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "left limit minus point value := sub(14, 1)",
      "after": "left limit minus point value = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先求左極限相對實際點值的差。",
        "en": "Compute the left limit minus the actual point value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left limit minus point value，不以小數近似替代。",
          "en": "Recompute left limit minus point value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "14",
          "1"
        ],
        "output": "13",
        "label": "left limit minus point value"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "right limit minus point value := sub(-12, 1)",
      "after": "right limit minus point value = -13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "右極限也必須與同一點值比較。",
        "en": "Compare the right limit with the same actual point value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 right limit minus point value，不以小數近似替代。",
          "en": "Recompute right limit minus point value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-12",
          "1"
        ],
        "output": "-13",
        "label": "right limit minus point value"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "L−f(a)=13;R−f(a)=-13",
      "after": "continuous at a=否",
      "relation": "implies",
      "rule_id": "two_sided_limit_equals_point",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在此兩側皆有仿射極限的實數模型，連續當且僅當左右極限與點值三者相等；只左右相等仍可能是補值錯誤。",
        "en": "In this real model with affine one-sided limits,continuity holds exactly when both limits and the point value agree.Equal left and right limits alone may still leave an incorrect center value."
      },
      "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": [
      "13",
      "-13"
    ],
    "left_limit": "14",
    "right_limit": "-12",
    "point_value": 1
  },
  "conclusion": {
    "canonical_value": "(13, -13)",
    "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-20261007-advance-20261008-r03-continuity-foundations",
      "batch_path": "public/data/questions/batches/batch-20261007-advance-20261008-r03-continuity-foundations/questions.jsonl",
      "batch_sha256": "1efeddd307a49cfd385a7e27892fe281bb5880419652b78ce889fdc4b0106df2",
      "original_line": 89,
      "original_line_sha256": "382152d2665477f4d7d3be4435c6c9f4a8f4137af556628a6b43db7841145f8d",
      "question_compact_sha256": "61b9dbfe8b9a1ccf89012d5b3488ccb06e741798951a3f40f385bb70a2d76c3f",
      "english_row_compact_sha256": "ad0a7f016614809dd3a67a6dc2cdc29acdbe4e63fd5a2387ada62c6ab50e1d09",
      "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261008-r03-continuity-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261008-r03-continuity-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-continuity": {
        "sha256": "500bf8bf13b82d10311338ec1e1ec96d1b7b1d3fcff6113697be491180bdb9e6",
        "english_sha256": "3d9b3ecf4edf7bc99e0a106ce8d24abe7e3fe8a8423ef9ab92a33510611f8d61",
        "original_path": "public/data/mko/continuity.json",
        "english_path": "public/data/i18n/editions/mko-continuity.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "66b95c054a23815e6aa9999f9605667abcb96e0c5a97b71cd67b67041065878a",
    "original_candidate_sha256": "a2f8746a50a89cfec27ca4463c3816cffca618c0ee715a35fce361baed228d46",
    "original_candidate_line_sha256": "a2f8746a50a89cfec27ca4463c3816cffca618c0ee715a35fce361baed228d46"
  },
  "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-20261026-004-p6",
    "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": "5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2",
      "question_first_witness_sha256": "1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 9,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "477268e970a5342c13f8ca1a4446cdedc6e37b7804ac2e7cbeaefe97699c76c1",
      "arithmetic_steps": 6
    },
    "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."
  }
}