{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261010-advance-20261014-r03-equation-0095",
  "pilot_number": "R0012-06620",
  "number": 6620,
  "family_id": "qf-density14-equation-common-shift-check",
  "title": {
    "zh-Hant": "第 6620 題：同量平移的雙側與殘差",
    "en": "Question 6620: Both Equation Sides after a Common Shift"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261010-advance-20261014-r03-equation-0095",
    "batch_id": "batch-20261010-advance-20261014-r03-equation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-equation"
    ],
    "title_zh": "同量平移的雙側與殘差",
    "stem_zh": "共同有理論域；依原式定義資格；L=1x+(-19)、R=-1x+(20)，x=19/2，同加e=0。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。",
    "answer": {
      "kind": "expression",
      "value": "(-20/1, -19/2, 21/2, -20/1, 否)",
      "explanation_zh": "同量平移雙側與殘差=(-20/1, -19/2, 21/2, -20/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 3,
      "skills": [
        "同量平移的雙側與殘差"
      ]
    },
    "variant": {
      "family_id": "qf-density14-equation-common-shift-check",
      "seed": 202610143095,
      "parameters": {
        "a": 1,
        "b": -19,
        "c": -1,
        "d": 20,
        "x": [
          19,
          2
        ],
        "e": 0,
        "domain": "finite_Q_original_equation_protocol",
        "operation": "density14_equation_common_shift_check",
        "semantic_template": "qf-density14-equation-common-shift-check/[\"-19/2\",\"21/2\"]"
      }
    },
    "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": "2026-10-10",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261010-advance-20261014-r03-equation-0095",
    "batch_id": "batch-20261010-advance-20261014-r03-equation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ff533b63876ebfd864b26a6159a9d24f6fd9c86f4a3996c1d4dccefb6d01aef7",
      "family_id": "qf-density14-equation-common-shift-check",
      "seed": 202610143095,
      "semantic_template": "qf-density14-equation-common-shift-check/[\"-19/2\",\"21/2\"]"
    },
    "fields": {
      "title": "Both Equation Sides after a Common Shift",
      "stem": "Over Q, L=(1)x+(-19), R=(-1)x+(20), x=19/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(-20/1, -19/2, 21/2, -20/1, No)",
        "explanation": "Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-20/1, -19/2, 21/2, -20/1, No)."
      },
      "skills": [
        "Both Equation Sides after a Common Shift"
      ],
      "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": "共同有理論域；依原式定義資格；L=1x+(-19)、R=-1x+(20)，x=19/2，同加e=0。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。",
        "en": "Over Q, L=(1)x+(-19), R=(-1)x+(20), x=19/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "common rational domain with original equation definitions",
      "after": "evaluate/classify the given model without enlarging its original domain",
      "relation": "implies",
      "rule_id": "original_rational_equation_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原式定義資格先於代入與變形；只核對明示模型，不把交乘或某點數值當成任意方程的一般求解證明。",
        "en": "Original definedness precedes substitution and transformations.Check the declared model,not arbitrary equation solving inferred from cross multiplication or one-point values."
      },
      "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 left numerator linear product := mul(1, 19/2)",
      "after": "original left numerator linear product = 19/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先把指定x代入此側係數乘積。",
        "en": "Substitute the specified x into this side's coefficient product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original left numerator linear product，不以小數近似替代。",
          "en": "Recompute original left numerator linear product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "19/2"
        ],
        "output": "19/2",
        "label": "original left numerator linear product"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "original left numerator affine value := sum(19/2, -19)",
      "after": "original left numerator affine value = -19/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此側自己的常數，不混用另一側。",
        "en": "Then add this side's own constant,not the other side's."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original left numerator affine value，不以小數近似替代。",
          "en": "Recompute original left numerator affine value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "19/2",
          "-19"
        ],
        "output": "-19/2",
        "label": "original left numerator affine value"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "original right linear product := mul(-1, 19/2)",
      "after": "original right linear product = -19/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先把指定x代入此側係數乘積。",
        "en": "Substitute the specified x into this side's coefficient product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original right linear product，不以小數近似替代。",
          "en": "Recompute original right linear product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "19/2"
        ],
        "output": "-19/2",
        "label": "original right linear product"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "original right affine value := sum(-19/2, 20)",
      "after": "original right affine value = 21/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此側自己的常數，不混用另一側。",
        "en": "Then add this side's own constant,not the other side's."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original right affine value，不以小數近似替代。",
          "en": "Recompute original right affine value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-19/2",
          "20"
        ],
        "output": "21/2",
        "label": "original right affine value"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "original L−R := sub(-19/2, 21/2)",
      "after": "original L−R = -20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先記原兩側差。",
        "en": "Record the original difference of the two sides first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original L−R，不以小數近似替代。",
          "en": "Recompute original L−R with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-19/2",
          "21/2"
        ],
        "output": "-20",
        "label": "original L−R"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "left plus common e := sum(-19/2, 0)",
      "after": "left plus common e = -19/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左側加同一已定有理量e。",
        "en": "Add the same defined rational e to the left."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left plus common e，不以小數近似替代。",
          "en": "Recompute left plus common e with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-19/2",
          "0"
        ],
        "output": "-19/2",
        "label": "left plus common e"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "right plus common e := sum(21/2, 0)",
      "after": "right plus common e = 21/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "右側也加同一e，不改為另一常數。",
        "en": "Add the identical e to the right,not another constant."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 right plus common e，不以小數近似替代。",
          "en": "Recompute right plus common e with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21/2",
          "0"
        ],
        "output": "21/2",
        "label": "right plus common e"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "shifted two-side residual := sub(-19/2, 21/2)",
      "after": "shifted two-side residual = -20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相減時兩個e抵消，應與原差一致。",
        "en": "Subtracting cancels the two copies of e and must recover the original difference."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 shifted two-side residual，不以小數近似替代。",
          "en": "Recompute shifted two-side residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-19/2",
          "21/2"
        ],
        "output": "-20",
        "label": "shifted two-side residual"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "original difference=-20;shifted difference=-20",
      "after": "shifted equality=否",
      "relation": "implies",
      "rule_id": "common_shift_reversible_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩側同加e可用兩側同減e逆轉，故在共同有理論域且都已定義時保留相等判準。",
        "en": "Adding e to both sides is reversed by subtracting e from both sides,so it preserves equality on the common defined rational domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "用相同e兩側同減，回到原式；不擴張原始定義域。",
          "en": "Subtract the same e from both sides to recover the original equation without enlarging its domain."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "-20",
      "-19/2",
      "21/2",
      "-20",
      false
    ]
  },
  "conclusion": {
    "canonical_value": "(-20/1, -19/2, 21/2, -20/1, 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": {
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    },
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    "original_candidate_sha256": "c7373f264159807555f66c848f2026ee62ccfd4b0f3472379df21bfb8b663429",
    "original_candidate_line_sha256": "c7373f264159807555f66c848f2026ee62ccfd4b0f3472379df21bfb8b663429"
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  "historical_source_state": {
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      "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-p13",
    "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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    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 10,
    "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."
  }
}