{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-r04-quadratic-equation-0082",
  "pilot_number": "R0055-28432",
  "number": 28432,
  "family_id": "qf-quadratic-exact-roots",
  "title": {
    "zh-Hant": "第 28432 題：二次方程式的精確根集合",
    "en": "Question 28432: Exact Root Set of a Quadratic Equation"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-r04-quadratic-equation-0082",
    "batch_id": "batch-20261003-r04-quadratic-equation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-quadratic-equation"
    ],
    "title_zh": "二次方程式的精確根集合",
    "stem_zh": "在實數論域，(6)x²+(12)x+(-18)=0 的 a≠0 且 D 為非負完全平方數。求最簡正分母、遞增且重複只列一次的根集合。",
    "answer": {
      "kind": "expression",
      "value": "{-3/1, 1/1}",
      "explanation_zh": "D=576；根={-3/1, 1/1}。用 (-b±√D)/(2a) 得精確有理根，再排序並合併重根；不用浮點比較。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 1,
      "skills": [
        "二次方程式的精確根集合"
      ]
    },
    "variant": {
      "family_id": "qf-quadratic-exact-roots",
      "seed": 202610034082,
      "parameters": {
        "a": 6,
        "b": 12,
        "c": -18,
        "domain": "reals",
        "operation": "exact_roots",
        "semantic_template": "qf-quadratic-exact-roots/{\"a\":6,\"b\":12,\"c\":-18,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication"
      ],
      "warning_zh": "僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex independent peer (inherited model; exact runtime model not exposed)",
      "generated_at": "2026-10-03",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261003-r04-quadratic-equation-0082",
    "batch_id": "batch-20261003-r04-quadratic-equation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "c9daef35591e0e6332c57c163db23c6ee830a311c215be7c5e04ca69819b673b",
      "family_id": "qf-quadratic-exact-roots",
      "seed": 202610034082,
      "semantic_template": "qf-quadratic-exact-roots/{\"a\":6,\"b\":12,\"c\":-18,\"domain\":\"reals\",\"operation\":\"exact_roots\"}"
    },
    "fields": {
      "title": "Exact Root Set of a Quadratic Equation",
      "stem": "Over the reals, (6)x²+(12)x+(-18)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.",
      "answer": {
        "kind": "expression",
        "value": "{-3/1, 1/1}",
        "explanation": "D=576. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-3/1, 1/1}. The comparison is exact."
      },
      "skills": [
        "Exact Root Set of a Quadratic Equation"
      ],
      "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": "在實數論域，(6)x²+(12)x+(-18)=0 的 a≠0 且 D 為非負完全平方數。求最簡正分母、遞增且重複只列一次的根集合。",
        "en": "Over the reals, (6)x²+(12)x+(-18)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-quadratic-exact-roots, exact_roots]",
      "after": "derived = [[increasing_reduced_positive_denominator_root_SET_retaining1]]",
      "relation": "implies",
      "rule_id": "source_answer_roles_or_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此頁明列本題scalar／label／set／point或mode欄序，不強套現代tuple。舊題domain與operation底線格式按原資料，分數整數亦保/1；原中英快照不改。",
        "en": "List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Legacy quadratic ORIGINAL b squared := mul(12, 12)",
      "after": "Legacy quadratic ORIGINAL b squared = 144",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic ORIGINAL b squared，不以小數近似替代。",
          "en": "Recompute Legacy quadratic ORIGINAL b squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "12"
        ],
        "output": "144",
        "label": "Legacy quadratic ORIGINAL b squared"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Legacy quadratic ORIGINAL a times c := mul(6, -18)",
      "after": "Legacy quadratic ORIGINAL a times c = -108",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic ORIGINAL a times c，不以小數近似替代。",
          "en": "Recompute Legacy quadratic ORIGINAL a times c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "-18"
        ],
        "output": "-108",
        "label": "Legacy quadratic ORIGINAL a times c"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Legacy quadratic ORIGINAL four times ac := mul(4, -108)",
      "after": "Legacy quadratic ORIGINAL four times ac = -432",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic ORIGINAL four times ac，不以小數近似替代。",
          "en": "Recompute Legacy quadratic ORIGINAL four times ac with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "-108"
        ],
        "output": "-432",
        "label": "Legacy quadratic ORIGINAL four times ac"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Legacy quadratic ORIGINAL b squared minus fourac := sub(144, -432)",
      "after": "Legacy quadratic ORIGINAL b squared minus fourac = 576",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic ORIGINAL b squared minus fourac，不以小數近似替代。",
          "en": "Recompute Legacy quadratic ORIGINAL b squared minus fourac with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "144",
          "-432"
        ],
        "output": "576",
        "label": "Legacy quadratic ORIGINAL b squared minus fourac"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [6, 12, -18, 576]",
      "after": "derived = [4a*(a*x^2+b*x+c)=(2a*x+b)^2-D, 2]",
      "relation": "implies",
      "rule_id": "Legacy_quadratic_GENERAL_complete_square_iff_DISTINCT_real_roots",
      "conditions": [],
      "explanation": {
        "zh-Hant": "a非0使4a可逆，方程等價(2ax+b)²=D。D<0無實根，D0一個值，D>0兩個相異值；重根只算一次，D為不設衍生SAFE上限的BigInt。",
        "en": "With a!=0,multiplication by4a is invertible and the equation is equivalent to(2ax+b)²=D.Negative D gives no real root,zero D one value and positive D two DISTINCT values.A repeated root counts once;D is unbounded derived BigInt."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Legacy quadratic DERIVED discriminant exact midpoint square0 := mul(288, 288)",
      "after": "Legacy quadratic DERIVED discriminant exact midpoint square0 = 82944",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic DERIVED discriminant exact midpoint square0，不以小數近似替代。",
          "en": "Recompute Legacy quadratic DERIVED discriminant exact midpoint square0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "288",
          "288"
        ],
        "output": "82944",
        "label": "Legacy quadratic DERIVED discriminant exact midpoint square0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "source roles = [Legacy quadratic DERIVED discriminant, 576, 0, [0, 577], 288]",
      "after": "derived = [82944, 否, 0, 288]",
      "relation": "implies",
      "rule_id": "Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在非負整數上square單調，保持low²<=n<high²；mid是BigInt floor((low+high)/2)，不用浮點sqrt，嚴格縮小整數bracket。衍生值不重套SAFE。",
        "en": "Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Legacy quadratic DERIVED discriminant exact midpoint square1 := mul(144, 144)",
      "after": "Legacy quadratic DERIVED discriminant exact midpoint square1 = 20736",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic DERIVED discriminant exact midpoint square1，不以小數近似替代。",
          "en": "Recompute Legacy quadratic DERIVED discriminant exact midpoint square1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "144",
          "144"
        ],
        "output": "20736",
        "label": "Legacy quadratic DERIVED discriminant exact midpoint square1"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "source roles = [Legacy quadratic DERIVED discriminant, 576, 1, [0, 288], 144]",
      "after": "derived = [20736, 否, 0, 144]",
      "relation": "implies",
      "rule_id": "Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在非負整數上square單調，保持low²<=n<high²；mid是BigInt floor((low+high)/2)，不用浮點sqrt，嚴格縮小整數bracket。衍生值不重套SAFE。",
        "en": "Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Legacy quadratic DERIVED discriminant exact midpoint square2 := mul(72, 72)",
      "after": "Legacy quadratic DERIVED discriminant exact midpoint square2 = 5184",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic DERIVED discriminant exact midpoint square2，不以小數近似替代。",
          "en": "Recompute Legacy quadratic DERIVED discriminant exact midpoint square2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "72",
          "72"
        ],
        "output": "5184",
        "label": "Legacy quadratic DERIVED discriminant exact midpoint square2"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "source roles = [Legacy quadratic DERIVED discriminant, 576, 2, [0, 144], 72]",
      "after": "derived = [5184, 否, 0, 72]",
      "relation": "implies",
      "rule_id": "Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在非負整數上square單調，保持low²<=n<high²；mid是BigInt floor((low+high)/2)，不用浮點sqrt，嚴格縮小整數bracket。衍生值不重套SAFE。",
        "en": "Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Legacy quadratic DERIVED discriminant exact midpoint square3 := mul(36, 36)",
      "after": "Legacy quadratic DERIVED discriminant exact midpoint square3 = 1296",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic DERIVED discriminant exact midpoint square3，不以小數近似替代。",
          "en": "Recompute Legacy quadratic DERIVED discriminant exact midpoint square3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "36",
          "36"
        ],
        "output": "1296",
        "label": "Legacy quadratic DERIVED discriminant exact midpoint square3"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "source roles = [Legacy quadratic DERIVED discriminant, 576, 3, [0, 72], 36]",
      "after": "derived = [1296, 否, 0, 36]",
      "relation": "implies",
      "rule_id": "Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在非負整數上square單調，保持low²<=n<high²；mid是BigInt floor((low+high)/2)，不用浮點sqrt，嚴格縮小整數bracket。衍生值不重套SAFE。",
        "en": "Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Legacy quadratic DERIVED discriminant exact midpoint square4 := mul(18, 18)",
      "after": "Legacy quadratic DERIVED discriminant exact midpoint square4 = 324",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic DERIVED discriminant exact midpoint square4，不以小數近似替代。",
          "en": "Recompute Legacy quadratic DERIVED discriminant exact midpoint square4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "18",
          "18"
        ],
        "output": "324",
        "label": "Legacy quadratic DERIVED discriminant exact midpoint square4"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "source roles = [Legacy quadratic DERIVED discriminant, 576, 4, [0, 36], 18]",
      "after": "derived = [324, 是, 18, 36]",
      "relation": "implies",
      "rule_id": "Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在非負整數上square單調，保持low²<=n<high²；mid是BigInt floor((low+high)/2)，不用浮點sqrt，嚴格縮小整數bracket。衍生值不重套SAFE。",
        "en": "Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Legacy quadratic DERIVED discriminant exact midpoint square5 := mul(27, 27)",
      "after": "Legacy quadratic DERIVED discriminant exact midpoint square5 = 729",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic DERIVED discriminant exact midpoint square5，不以小數近似替代。",
          "en": "Recompute Legacy quadratic DERIVED discriminant exact midpoint square5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "27",
          "27"
        ],
        "output": "729",
        "label": "Legacy quadratic DERIVED discriminant exact midpoint square5"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "source roles = [Legacy quadratic DERIVED discriminant, 576, 5, [18, 36], 27]",
      "after": "derived = [729, 否, 18, 27]",
      "relation": "implies",
      "rule_id": "Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在非負整數上square單調，保持low²<=n<high²；mid是BigInt floor((low+high)/2)，不用浮點sqrt，嚴格縮小整數bracket。衍生值不重套SAFE。",
        "en": "Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Legacy quadratic DERIVED discriminant exact midpoint square6 := mul(22, 22)",
      "after": "Legacy quadratic DERIVED discriminant exact midpoint square6 = 484",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic DERIVED discriminant exact midpoint square6，不以小數近似替代。",
          "en": "Recompute Legacy quadratic DERIVED discriminant exact midpoint square6 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "22"
        ],
        "output": "484",
        "label": "Legacy quadratic DERIVED discriminant exact midpoint square6"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "source roles = [Legacy quadratic DERIVED discriminant, 576, 6, [18, 27], 22]",
      "after": "derived = [484, 是, 22, 27]",
      "relation": "implies",
      "rule_id": "Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在非負整數上square單調，保持low²<=n<high²；mid是BigInt floor((low+high)/2)，不用浮點sqrt，嚴格縮小整數bracket。衍生值不重套SAFE。",
        "en": "Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Legacy quadratic DERIVED discriminant exact midpoint square7 := mul(24, 24)",
      "after": "Legacy quadratic DERIVED discriminant exact midpoint square7 = 576",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic DERIVED discriminant exact midpoint square7，不以小數近似替代。",
          "en": "Recompute Legacy quadratic DERIVED discriminant exact midpoint square7 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "24"
        ],
        "output": "576",
        "label": "Legacy quadratic DERIVED discriminant exact midpoint square7"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "source roles = [Legacy quadratic DERIVED discriminant, 576, 7, [22, 27], 24]",
      "after": "derived = [576, 是, 24, 27]",
      "relation": "implies",
      "rule_id": "Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在非負整數上square單調，保持low²<=n<high²；mid是BigInt floor((low+high)/2)，不用浮點sqrt，嚴格縮小整數bracket。衍生值不重套SAFE。",
        "en": "Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Legacy quadratic DERIVED discriminant exact midpoint square8 := mul(25, 25)",
      "after": "Legacy quadratic DERIVED discriminant exact midpoint square8 = 625",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic DERIVED discriminant exact midpoint square8，不以小數近似替代。",
          "en": "Recompute Legacy quadratic DERIVED discriminant exact midpoint square8 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "25"
        ],
        "output": "625",
        "label": "Legacy quadratic DERIVED discriminant exact midpoint square8"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "source roles = [Legacy quadratic DERIVED discriminant, 576, 8, [24, 27], 25]",
      "after": "derived = [625, 否, 24, 25]",
      "relation": "implies",
      "rule_id": "Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在非負整數上square單調，保持low²<=n<high²；mid是BigInt floor((low+high)/2)，不用浮點sqrt，嚴格縮小整數bracket。衍生值不重套SAFE。",
        "en": "Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Legacy quadratic DERIVED discriminant terminal root square := mul(24, 24)",
      "after": "Legacy quadratic DERIVED discriminant terminal root square = 576",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic DERIVED discriminant terminal root square，不以小數近似替代。",
          "en": "Recompute Legacy quadratic DERIVED discriminant terminal root square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "24"
        ],
        "output": "576",
        "label": "Legacy quadratic DERIVED discriminant terminal root square"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Legacy quadratic DERIVED discriminant terminal successor square := mul(25, 25)",
      "after": "Legacy quadratic DERIVED discriminant terminal successor square = 625",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic DERIVED discriminant terminal successor square，不以小數近似替代。",
          "en": "Recompute Legacy quadratic DERIVED discriminant terminal successor square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "25"
        ],
        "output": "625",
        "label": "Legacy quadratic DERIVED discriminant terminal successor square"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "source roles = [Legacy quadratic DERIVED discriminant, 576, 24, 25]",
      "after": "derived = [576, 625]",
      "relation": "implies",
      "rule_id": "Legacy_isqrt_terminal_floor_and_single_nonnegative_branch",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相鄰整數bracket證floor sqrt=low；exact root另核low²=n，主平方根只取非負分支。",
        "en": "Adjacent integer bounds certify floor sqrt=low;an exact root separately needs low²=n,and the principal root is the single nonnegative branch."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Legacy quadratic ORIGINAL negated b := sub(0, 12)",
      "after": "Legacy quadratic ORIGINAL negated b = -12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic ORIGINAL negated b，不以小數近似替代。",
          "en": "Recompute Legacy quadratic ORIGINAL negated b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "12"
        ],
        "output": "-12",
        "label": "Legacy quadratic ORIGINAL negated b"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Legacy quadratic ORIGINAL minus numerator := sub(-12, 24)",
      "after": "Legacy quadratic ORIGINAL minus numerator = -36",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic ORIGINAL minus numerator，不以小數近似替代。",
          "en": "Recompute Legacy quadratic ORIGINAL minus numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-12",
          "24"
        ],
        "output": "-36",
        "label": "Legacy quadratic ORIGINAL minus numerator"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Legacy quadratic ORIGINAL plus numerator := sum(-12, 24)",
      "after": "Legacy quadratic ORIGINAL plus numerator = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic ORIGINAL plus numerator，不以小數近似替代。",
          "en": "Recompute Legacy quadratic ORIGINAL plus numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-12",
          "24"
        ],
        "output": "12",
        "label": "Legacy quadratic ORIGINAL plus numerator"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Legacy quadratic ORIGINAL denominator2a := mul(2, 6)",
      "after": "Legacy quadratic ORIGINAL denominator2a = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic ORIGINAL denominator2a，不以小數近似替代。",
          "en": "Recompute Legacy quadratic ORIGINAL denominator2a with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "6"
        ],
        "output": "12",
        "label": "Legacy quadratic ORIGINAL denominator2a"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "Legacy quadratic ORIGINAL raw rational branch0 := div(-36, 12)",
      "after": "Legacy quadratic ORIGINAL raw rational branch0 = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic ORIGINAL raw rational branch0，不以小數近似替代。",
          "en": "Recompute Legacy quadratic ORIGINAL raw rational branch0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-36",
          "12"
        ],
        "output": "-3",
        "label": "Legacy quadratic ORIGINAL raw rational branch0"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "source roles = [0, -36, 12]",
      "after": "derived = [-3, 1, -3/1]",
      "relation": "implies",
      "rule_id": "Legacy_quadratic_EACH_raw_branch_positive_den_gcd_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個原branch normalize分母正號與gcd；原den2a可負，衍生num／den不重套SAFE，整數仍顯示p/1。",
        "en": "Normalize positive denominator and gcd for EACH original branch.Original2a may be negative;derived numerator/denominator do not inherit SAFE.Integer roots retain p/1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "Legacy quadratic ORIGINAL raw rational branch1 := div(12, 12)",
      "after": "Legacy quadratic ORIGINAL raw rational branch1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy quadratic ORIGINAL raw rational branch1，不以小數近似替代。",
          "en": "Recompute Legacy quadratic ORIGINAL raw rational branch1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "12",
          "12"
        ],
        "output": "1",
        "label": "Legacy quadratic ORIGINAL raw rational branch1"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "source roles = [1, 12, 12]",
      "after": "derived = [1, 1, 1/1]",
      "relation": "implies",
      "rule_id": "Legacy_quadratic_EACH_raw_branch_positive_den_gcd_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個原branch normalize分母正號與gcd；原den2a可負，衍生num／den不重套SAFE，整數仍顯示p/1。",
        "en": "Normalize positive denominator and gcd for EACH original branch.Original2a may be negative;derived numerator/denominator do not inherit SAFE.Integer roots retain p/1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "source roles = [6, 576, [-3/1, 1/1]]",
      "after": "derived = [[-3/1, 1/1], {-3/1, 1/1}]",
      "relation": "implies",
      "rule_id": "Legacy_quadratic_EXACT_numeric_sort_and_D0_set_dedup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按有理數實值／cross products排序，不能lexical或假設minus branch一定較小（a負會反向）。D0的兩個branch同值只列一次；完整square iff也證沒有漏根。",
        "en": "Sort by exact rational value/cross products,not lexically or by assuming the minus branch is smaller(negative a reverses it).At D0 list the repeated value once;the complete-square IFF also proves no roots omitted."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "{-3/1, 1/1}"
  },
  "conclusion": {
    "canonical_value": "{-3/1, 1/1}",
    "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": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261003-r04-quadratic-equation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261003-r04-quadratic-equation-foundations/questions.jsonl",
      "batch_sha256": "66a176b962ca84e02bebfdfca16c81ee9677788154f1765c2bfa965a41fce5f6",
      "original_line": 82,
      "original_line_sha256": "764917b386a9eba17585fa12efcfa105486dc134a43b8381134599073017cd25",
      "question_compact_sha256": "c9daef35591e0e6332c57c163db23c6ee830a311c215be7c5e04ca69819b673b",
      "english_row_compact_sha256": "f2ac2b10b216ecc99c8bedacc6ef9f0ecf9e9d9b278454be91779f8d496652ca",
      "english_manifest_path": "public/data/i18n/editions/batch-20261003-r04-quadratic-equation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261003-r04-quadratic-equation-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-quadratic-equation": {
        "sha256": "d04aa30d9e57bfe7e7df1f57f60f8361ce982217aa4548374680659f56b38851",
        "english_sha256": "733cbb98c4349ec66f14ef06f4f713641d49d16ee06b555538ec686c7756deaa",
        "original_path": "public/data/mko/quadratic-equation.json",
        "english_path": "public/data/i18n/editions/mko-quadratic-equation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b5763df142e86ef7f9787b6ea86050afc60bd3c5c53538d4839606827cc44686",
    "original_candidate_sha256": "1a3b9ed96553551410453ff12e013c852d0228104e5e951ff67e6cd9244c1ba8",
    "original_candidate_line_sha256": "1a3b9ed96553551410453ff12e013c852d0228104e5e951ff67e6cd9244c1ba8"
  },
  "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-20261026-010-p56",
    "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": "6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683",
      "question_first_witness_sha256": "611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 36,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "c7494c32b154df7205ebf5e8994a09d61fd0e9fbf27234293b711a5b7ccfb29e",
      "arithmetic_steps": 21
    },
    "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."
  }
}