{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261009-r03-function-limit-0070",
  "pilot_number": "R0006-03970",
  "number": 3970,
  "family_id": "qf-density-limit-one-sided-pair",
  "title": {
    "zh-Hant": "第 3970 題：左右去心極限的有序對",
    "en": "Question 3970: Ordered Left and Right Punctured Limits"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261009-r03-function-limit-0070",
    "batch_id": "batch-20261007-advance-20261009-r03-function-limit-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-function-limit"
    ],
    "title_zh": "左右去心極限的有序對",
    "stem_zh": "實數模型；點值與去心極限分別解讀。a=1；x<a時 f(x)=2x+(5)，x>a時 f(x)=-2x+(-4)。列 (左極限, 右極限)，不交換順序。",
    "answer": {
      "kind": "expression",
      "value": "(7, -6)",
      "explanation_zh": "左右極限=(7, -6)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "左右去心極限的有序對"
      ]
    },
    "variant": {
      "family_id": "qf-density-limit-one-sided-pair",
      "seed": 202610093070,
      "parameters": {
        "a": 1,
        "left_m": 2,
        "left_b": 5,
        "right_m": -2,
        "right_b": -4,
        "operation": "density_limit_one_sided_pair",
        "domain": "punctured_reals",
        "semantic_template": "qf-density-limit-one-sided-pair/[\"7\",\"-6\"]"
      }
    },
    "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-20261009-r03-function-limit-0070",
    "batch_id": "batch-20261007-advance-20261009-r03-function-limit-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "fb64d0485585f9a96b69fcd6ee2730cbea67038297c8651d7d6da2f9b2d74a7e",
      "family_id": "qf-density-limit-one-sided-pair",
      "seed": 202610093070,
      "semantic_template": "qf-density-limit-one-sided-pair/[\"7\",\"-6\"]"
    },
    "fields": {
      "title": "Ordered Left and Right Punctured Limits",
      "stem": "On R∖{1}, the branch for x<a is (2)x+(5), and for x>a is (-2)x+(-4). Give (left limit,right limit), keeping the order.",
      "answer": {
        "kind": "expression",
        "value": "(7, -6)",
        "explanation": "Evaluate the two affine approach values to obtain (7, -6). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."
      },
      "skills": [
        "Ordered Left and Right Punctured Limits"
      ],
      "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": "實數模型；點值與去心極限分別解讀。a=1；x<a時 f(x)=2x+(5)，x>a時 f(x)=-2x+(-4)。列 (左極限, 右極限)，不交換順序。",
        "en": "On R∖{1}, the branch for x<a is (2)x+(5), and for x>a is (-2)x+(-4). Give (left limit,right limit), keeping the order."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared real model with an excluded center for limit data",
      "after": "point value,legal off-center evaluation and limit are distinct",
      "relation": "implies",
      "rule_id": "punctured_limit_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "去心極限不讀中心點值；指定合法點的求值也不冒充極限。這些明示仿射或可約多項式模型不代表任意函數。",
        "en": "A punctured limit does not read the center value,and evaluation at one legal point is not a limit.These declared affine or factorizable models do not represent arbitrary functions."
      },
      "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": "取宣告分支係數乘中心。",
        "en": "Multiply the branch slope by the center."
      },
      "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 limit value := sum(2, 5)",
      "after": "left branch limit value = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加該分支常數；仿射差m(x−a)可用ε−δ控制。",
        "en": "Add that branch's constant;the affine difference m(x−a) admits ε−δ control."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left branch limit value，不以小數近似替代。",
          "en": "Recompute left branch limit value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "5"
        ],
        "output": "7",
        "label": "left branch limit 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": "取宣告分支係數乘中心。",
        "en": "Multiply the branch slope by the center."
      },
      "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 limit value := sum(-2, -4)",
      "after": "right branch limit value = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加該分支常數；仿射差m(x−a)可用ε−δ控制。",
        "en": "Add that branch's constant;the affine difference m(x−a) admits ε−δ control."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 right branch limit value，不以小數近似替代。",
          "en": "Recompute right branch limit value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "-4"
        ],
        "output": "-6",
        "label": "right branch limit value"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "left limit=7;right limit=-6",
      "after": "ordered one-sided pair=(7,-6)",
      "relation": "implies",
      "rule_id": "one_sided_affine_limit_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每側的仿射差各自用δ=ε/(1+|m|)控制。兩側相同才有共同雙側極限；不同則無雙側極限，本題都不補中心點值。",
        "en": "Control each affine side with δ=ε/(1+|m|).Equal side limits give a common two-sided limit;unequal limits do not.Neither case fills a 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": [
      "7",
      "-6"
    ]
  },
  "conclusion": {
    "canonical_value": "(7, -6)",
    "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-20261009-r03-function-limit-foundations",
      "batch_path": "public/data/questions/batches/batch-20261007-advance-20261009-r03-function-limit-foundations/questions.jsonl",
      "batch_sha256": "b4e71b191c5c0c73b5a782d764be0c2938b563b18f840324154a28dda4ced9fa",
      "original_line": 70,
      "original_line_sha256": "3dde34cc6017c750d6ab9c15447004e3a860c91e7ca9bf2d22f35b15fa753e20",
      "question_compact_sha256": "fb64d0485585f9a96b69fcd6ee2730cbea67038297c8651d7d6da2f9b2d74a7e",
      "english_row_compact_sha256": "469b69ecfef97c309405295f621d0f4509d3d164885b3f5756d15790747a6cca",
      "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261009-r03-function-limit-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261009-r03-function-limit-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-function-limit": {
        "sha256": "2dce49b3fcec9f9cf28bc85107ed2884a57b36ccc297c60e4ef5836b4a3c5ca3",
        "english_sha256": "608f319ceb9afeb2eb284c43c4a1d6d8d19c4e40dce81f0ac4df60cbb55f4c44",
        "original_path": "public/data/mko/function-limit.json",
        "english_path": "public/data/i18n/editions/mko-function-limit.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6bac5f83f28cdd9c992a469306bfdf5c5415de17d11847cb9b17b4d13934c7e6",
    "original_candidate_sha256": "0979e869502c15723479657f2e9c1b147392fa81d0640b203b5dcf2cf4a897e6",
    "original_candidate_line_sha256": "0979e869502c15723479657f2e9c1b147392fa81d0640b203b5dcf2cf4a897e6"
  },
  "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-p7",
    "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": "568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d",
      "question_first_witness_sha256": "d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "325501a02492872b678fecd5e2b287fa3b5a60f0b2593b81e65626b8d6056f03",
      "arithmetic_steps": 4
    },
    "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."
  }
}