{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261004-advance-20261005-r07-variance-0057",
  "pilot_number": "R0070-35807",
  "number": 35807,
  "family_id": "qf-variance-scale",
  "title": {
    "zh-Hant": "第 35807 題：縮放後的機率變異數",
    "en": "Question 35807: Probability Variance after Scaling"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261004-advance-20261005-r07-variance-0057",
    "batch_id": "batch-20261004-advance-20261005-r07-variance-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-variance"
    ],
    "title_zh": "縮放後的機率變異數",
    "stem_zh": "非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -4), (1, -4), (2, 4), (3, 1005)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/14, 8/14, 5/14, 0/14]，權重總和恰為一。令 Z=2X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。",
    "answer": {
      "kind": "expression",
      "value": "2880/49",
      "explanation_zh": "變異數=2880/49。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "縮放後的機率變異數"
      ]
    },
    "variant": {
      "family_id": "qf-variance-scale",
      "seed": 202610057057,
      "parameters": {
        "S": [
          0,
          1,
          2,
          3
        ],
        "F": "powerset(S)",
        "values": [
          -4,
          -4,
          4,
          1005
        ],
        "weights": [
          1,
          8,
          5,
          0
        ],
        "denominator": 14,
        "k": 2,
        "operation": "scale_variance",
        "semantic_template": "qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":2,\"law\":[[\"0\",\"9/14\"],[\"8\",\"5/14\"]],\"operation\":\"scale_variance\"}"
      }
    },
    "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": "Codex independent producer (inherited model; exact runtime model not exposed)",
      "generated_at": "2026-10-04",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261004-advance-20261005-r07-variance-0057",
    "batch_id": "batch-20261004-advance-20261005-r07-variance-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ca242f0cdd9e5a183eeafeb68d0701593508f836b7bd4ffdc7faa15b714b286b",
      "family_id": "qf-variance-scale",
      "seed": 202610057057,
      "semantic_template": "qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":2,\"law\":[[\"0\",\"9/14\"],[\"8\",\"5/14\"]],\"operation\":\"scale_variance\"}"
    },
    "fields": {
      "title": "Probability Variance after Scaling",
      "stem": "The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -4), (1, -4), (2, 4), (3, 1005)}. In the displayed order of S, the nonnegative probability weights are [1/(14), 8/(14), 5/(14), 0/(14)], summing exactly to one. Put Z=(2)X. Find Var(Z)=E[(Z−E[Z])²] exactly as a reduced positive-denominator fraction. This is probability-distribution variance, not sample variance with an n−1 denominator.",
      "answer": {
        "kind": "expression",
        "value": "2880/49",
        "explanation": "Compute the mean, then sum weighted squared deviations exactly to obtain 2880/49. Translation preserves deviations; scaling multiplies variance by k², including zero or negative k. Zero-weight outcomes contribute nothing. This is probability-distribution variance, not sample variance with an n−1 denominator."
      },
      "skills": [
        "Probability Variance after Scaling"
      ],
      "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": "非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -4), (1, -4), (2, 4), (3, 1005)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/14, 8/14, 5/14, 0/14]，權重總和恰為一。令 Z=2X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。",
        "en": "The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -4), (1, -4), (2, 4), (3, 1005)}. In the displayed order of S, the nonnegative probability weights are [1/(14), 8/(14), 5/(14), 0/(14)], summing exactly to one. Put Z=(2)X. Find Var(Z)=E[(Z−E[Z])²] exactly as a reduced positive-denominator fraction. This is probability-distribution variance, not sample variance with an n−1 denominator."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-variance-scale, [F, S, denominator, k, operation, values, weights]]",
      "after": "derived = [variance, scale_variance, null]",
      "relation": "implies",
      "rule_id": "R70_SOURCE_bank_specific_interface",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只使用此bank真正明示的欄位、operation/domain與型別；BASE容許保留額外metadata，不把LDL SAFE/集合上限或其他bank的去重規則搬來。",
        "en": "Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。",
          "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Variance ORIGINAL nonnegative RAW weights sum := sum(1, 8, 5, 0)",
      "after": "Variance ORIGINAL nonnegative RAW weights sum = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance ORIGINAL nonnegative RAW weights sum，不以小數近似替代。",
          "en": "Recompute Variance ORIGINAL nonnegative RAW weights sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "8",
          "5",
          "0"
        ],
        "output": "14",
        "label": "Variance ORIGINAL nonnegative RAW weights sum"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [[0, 1, 2, 3], powerset(S), [-4, -4, 4, 1005], [1, 8, 5, 0], 14, 2, null]",
      "after": "derived = [是]",
      "relation": "implies",
      "rule_id": "Variance_WHOLE_normalized_model_ORIGINAL_slot_alignment",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全raw值/權重即使零權重也先驗SAFE，S不重複且非空，F為完整powerset，權重非負和=q>0。值與權重沿原S槽位對齊，不獨立排序；c/k也先驗raw SAFE。",
        "en": "Type ALL RAW values/weights EVEN at zero weight.S is nonempty/distinct,F is powerset(S),weights are nonnegative and sum to q>0.Keep values/weights aligned to ORIGINAL S slots;do not sort independently.Type RAW c/k as well."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。",
          "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Variance transformed ORIGINAL value i=0 := mul(2, -4)",
      "after": "Variance transformed ORIGINAL value i=0 = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance transformed ORIGINAL value i=0，不以小數近似替代。",
          "en": "Recompute Variance transformed ORIGINAL value i=0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-4"
        ],
        "output": "-8",
        "label": "Variance transformed ORIGINAL value i=0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Variance weight times transformed value i=0 := mul(1, -8)",
      "after": "Variance weight times transformed value i=0 = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance weight times transformed value i=0，不以小數近似替代。",
          "en": "Recompute Variance weight times transformed value i=0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-8"
        ],
        "output": "-8",
        "label": "Variance weight times transformed value i=0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [0, 0, -4, 1, 2, null]",
      "after": "derived = [-8, 1/14]",
      "relation": "implies",
      "rule_id": "Variance_EACH_slot_transformation_preserves_probability",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原槽做Y=X、X+c或kX，權重機率仍w/q；衍生值可超SAFE，不能重當raw輸入截斷。零權重槽仍保持來源但不貢獻。",
        "en": "For EACH ORIGINAL slot,set Y=X,X+c or kX while probability remains w/q.Derived values may exceed SAFE and are not recapped as RAW.Zero-weight slots remain in the source but contribute nothing."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。",
          "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Variance transformed ORIGINAL value i=1 := mul(2, -4)",
      "after": "Variance transformed ORIGINAL value i=1 = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance transformed ORIGINAL value i=1，不以小數近似替代。",
          "en": "Recompute Variance transformed ORIGINAL value i=1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-4"
        ],
        "output": "-8",
        "label": "Variance transformed ORIGINAL value i=1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Variance weight times transformed value i=1 := mul(8, -8)",
      "after": "Variance weight times transformed value i=1 = -64",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance weight times transformed value i=1，不以小數近似替代。",
          "en": "Recompute Variance weight times transformed value i=1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "-8"
        ],
        "output": "-64",
        "label": "Variance weight times transformed value i=1"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "source roles = [1, 1, -4, 8, 2, null]",
      "after": "derived = [-8, 4/7]",
      "relation": "implies",
      "rule_id": "Variance_EACH_slot_transformation_preserves_probability",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原槽做Y=X、X+c或kX，權重機率仍w/q；衍生值可超SAFE，不能重當raw輸入截斷。零權重槽仍保持來源但不貢獻。",
        "en": "For EACH ORIGINAL slot,set Y=X,X+c or kX while probability remains w/q.Derived values may exceed SAFE and are not recapped as RAW.Zero-weight slots remain in the source but contribute nothing."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。",
          "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Variance transformed ORIGINAL value i=2 := mul(2, 4)",
      "after": "Variance transformed ORIGINAL value i=2 = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance transformed ORIGINAL value i=2，不以小數近似替代。",
          "en": "Recompute Variance transformed ORIGINAL value i=2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "4"
        ],
        "output": "8",
        "label": "Variance transformed ORIGINAL value i=2"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Variance weight times transformed value i=2 := mul(5, 8)",
      "after": "Variance weight times transformed value i=2 = 40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance weight times transformed value i=2，不以小數近似替代。",
          "en": "Recompute Variance weight times transformed value i=2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "8"
        ],
        "output": "40",
        "label": "Variance weight times transformed value i=2"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "source roles = [2, 2, 4, 5, 2, null]",
      "after": "derived = [8, 5/14]",
      "relation": "implies",
      "rule_id": "Variance_EACH_slot_transformation_preserves_probability",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原槽做Y=X、X+c或kX，權重機率仍w/q；衍生值可超SAFE，不能重當raw輸入截斷。零權重槽仍保持來源但不貢獻。",
        "en": "For EACH ORIGINAL slot,set Y=X,X+c or kX while probability remains w/q.Derived values may exceed SAFE and are not recapped as RAW.Zero-weight slots remain in the source but contribute nothing."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。",
          "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Variance transformed ORIGINAL value i=3 := mul(2, 1005)",
      "after": "Variance transformed ORIGINAL value i=3 = 2010",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance transformed ORIGINAL value i=3，不以小數近似替代。",
          "en": "Recompute Variance transformed ORIGINAL value i=3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1005"
        ],
        "output": "2010",
        "label": "Variance transformed ORIGINAL value i=3"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Variance weight times transformed value i=3 := mul(0, 2010)",
      "after": "Variance weight times transformed value i=3 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance weight times transformed value i=3，不以小數近似替代。",
          "en": "Recompute Variance weight times transformed value i=3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2010"
        ],
        "output": "0",
        "label": "Variance weight times transformed value i=3"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "source roles = [3, 3, 1005, 0, 2, null]",
      "after": "derived = [2010, 0]",
      "relation": "implies",
      "rule_id": "Variance_EACH_slot_transformation_preserves_probability",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原槽做Y=X、X+c或kX，權重機率仍w/q；衍生值可超SAFE，不能重當raw輸入截斷。零權重槽仍保持來源但不貢獻。",
        "en": "For EACH ORIGINAL slot,set Y=X,X+c or kX while probability remains w/q.Derived values may exceed SAFE and are not recapped as RAW.Zero-weight slots remain in the source but contribute nothing."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。",
          "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Variance mean numerator sum wY := sum(-8, -64, 40, 0)",
      "after": "Variance mean numerator sum wY = -32",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance mean numerator sum wY，不以小數近似替代。",
          "en": "Recompute Variance mean numerator sum wY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-8",
          "-64",
          "40",
          "0"
        ],
        "output": "-32",
        "label": "Variance mean numerator sum wY"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Variance exact expectation M over q := div(-32, 14)",
      "after": "Variance exact expectation M over q = -16/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance exact expectation M over q，不以小數近似替代。",
          "en": "Recompute Variance exact expectation M over q with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-32",
          "14"
        ],
        "output": "-16/7",
        "label": "Variance exact expectation M over q"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Variance i=0 Y times q := mul(-8, 14)",
      "after": "Variance i=0 Y times q = -112",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=0 Y times q，不以小數近似替代。",
          "en": "Recompute Variance i=0 Y times q with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-8",
          "14"
        ],
        "output": "-112",
        "label": "Variance i=0 Y times q"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Variance i=0 centered integer numerator := sub(-112, -32)",
      "after": "Variance i=0 centered integer numerator = -80",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=0 centered integer numerator，不以小數近似替代。",
          "en": "Recompute Variance i=0 centered integer numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-112",
          "-32"
        ],
        "output": "-80",
        "label": "Variance i=0 centered integer numerator"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Variance i=0 centered numerator squared := mul(-80, -80)",
      "after": "Variance i=0 centered numerator squared = 6400",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=0 centered numerator squared，不以小數近似替代。",
          "en": "Recompute Variance i=0 centered numerator squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-80",
          "-80"
        ],
        "output": "6400",
        "label": "Variance i=0 centered numerator squared"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Variance i=0 weight times squared numerator := mul(1, 6400)",
      "after": "Variance i=0 weight times squared numerator = 6400",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=0 weight times squared numerator，不以小數近似替代。",
          "en": "Recompute Variance i=0 weight times squared numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "6400"
        ],
        "output": "6400",
        "label": "Variance i=0 weight times squared numerator"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Variance i=1 Y times q := mul(-8, 14)",
      "after": "Variance i=1 Y times q = -112",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=1 Y times q，不以小數近似替代。",
          "en": "Recompute Variance i=1 Y times q with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-8",
          "14"
        ],
        "output": "-112",
        "label": "Variance i=1 Y times q"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Variance i=1 centered integer numerator := sub(-112, -32)",
      "after": "Variance i=1 centered integer numerator = -80",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=1 centered integer numerator，不以小數近似替代。",
          "en": "Recompute Variance i=1 centered integer numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-112",
          "-32"
        ],
        "output": "-80",
        "label": "Variance i=1 centered integer numerator"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Variance i=1 centered numerator squared := mul(-80, -80)",
      "after": "Variance i=1 centered numerator squared = 6400",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=1 centered numerator squared，不以小數近似替代。",
          "en": "Recompute Variance i=1 centered numerator squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-80",
          "-80"
        ],
        "output": "6400",
        "label": "Variance i=1 centered numerator squared"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Variance i=1 weight times squared numerator := mul(8, 6400)",
      "after": "Variance i=1 weight times squared numerator = 51200",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=1 weight times squared numerator，不以小數近似替代。",
          "en": "Recompute Variance i=1 weight times squared numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "6400"
        ],
        "output": "51200",
        "label": "Variance i=1 weight times squared numerator"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Variance i=2 Y times q := mul(8, 14)",
      "after": "Variance i=2 Y times q = 112",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=2 Y times q，不以小數近似替代。",
          "en": "Recompute Variance i=2 Y times q with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "14"
        ],
        "output": "112",
        "label": "Variance i=2 Y times q"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Variance i=2 centered integer numerator := sub(112, -32)",
      "after": "Variance i=2 centered integer numerator = 144",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=2 centered integer numerator，不以小數近似替代。",
          "en": "Recompute Variance i=2 centered integer numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "112",
          "-32"
        ],
        "output": "144",
        "label": "Variance i=2 centered integer numerator"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Variance i=2 centered numerator squared := mul(144, 144)",
      "after": "Variance i=2 centered numerator squared = 20736",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=2 centered numerator squared，不以小數近似替代。",
          "en": "Recompute Variance i=2 centered numerator squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "144",
          "144"
        ],
        "output": "20736",
        "label": "Variance i=2 centered numerator squared"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Variance i=2 weight times squared numerator := mul(5, 20736)",
      "after": "Variance i=2 weight times squared numerator = 103680",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=2 weight times squared numerator，不以小數近似替代。",
          "en": "Recompute Variance i=2 weight times squared numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "20736"
        ],
        "output": "103680",
        "label": "Variance i=2 weight times squared numerator"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Variance i=3 Y times q := mul(2010, 14)",
      "after": "Variance i=3 Y times q = 28140",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=3 Y times q，不以小數近似替代。",
          "en": "Recompute Variance i=3 Y times q with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2010",
          "14"
        ],
        "output": "28140",
        "label": "Variance i=3 Y times q"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Variance i=3 centered integer numerator := sub(28140, -32)",
      "after": "Variance i=3 centered integer numerator = 28172",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=3 centered integer numerator，不以小數近似替代。",
          "en": "Recompute Variance i=3 centered integer numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "28140",
          "-32"
        ],
        "output": "28172",
        "label": "Variance i=3 centered integer numerator"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "Variance i=3 centered numerator squared := mul(28172, 28172)",
      "after": "Variance i=3 centered numerator squared = 793661584",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=3 centered numerator squared，不以小數近似替代。",
          "en": "Recompute Variance i=3 centered numerator squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "28172",
          "28172"
        ],
        "output": "793661584",
        "label": "Variance i=3 centered numerator squared"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Variance i=3 weight times squared numerator := mul(0, 793661584)",
      "after": "Variance i=3 weight times squared numerator = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance i=3 weight times squared numerator，不以小數近似替代。",
          "en": "Recompute Variance i=3 weight times squared numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "793661584"
        ],
        "output": "0",
        "label": "Variance i=3 weight times squared numerator"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "Variance sum ALL weighted squared numerators := sum(6400, 51200, 103680, 0)",
      "after": "Variance sum ALL weighted squared numerators = 161280",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance sum ALL weighted squared numerators，不以小數近似替代。",
          "en": "Recompute Variance sum ALL weighted squared numerators with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6400",
          "51200",
          "103680",
          "0"
        ],
        "output": "161280",
        "label": "Variance sum ALL weighted squared numerators"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "Variance denominator q squared := mul(14, 14)",
      "after": "Variance denominator q squared = 196",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance denominator q squared，不以小數近似替代。",
          "en": "Recompute Variance denominator q squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "14"
        ],
        "output": "196",
        "label": "Variance denominator q squared"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "Variance denominator q cubed binary second factor := mul(196, 14)",
      "after": "Variance denominator q cubed binary second factor = 2744",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance denominator q cubed binary second factor，不以小數近似替代。",
          "en": "Recompute Variance denominator q cubed binary second factor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "196",
          "14"
        ],
        "output": "2744",
        "label": "Variance denominator q cubed binary second factor"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Variance probability population centered sum over q cubed := div(161280, 2744)",
      "after": "Variance probability population centered sum over q cubed = 2880/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。",
        "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Variance probability population centered sum over q cubed，不以小數近似替代。",
          "en": "Recompute Variance probability population centered sum over q cubed with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "161280",
          "2744"
        ],
        "output": "2880/49",
        "label": "Variance probability population centered sum over q cubed"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "source roles = [[-8, -8, 8, 2010], [1, 8, 5, 0], 14, -16/7, 161280, 2744]",
      "after": "derived = [2880/49, [-8, 8], 否]",
      "relation": "implies",
      "rule_id": "Variance_CENTERED_definition_and_positive_support_zero_equivalence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "E[Y]=M/q，Var=Σ(w/q)(Y−M/q)²=C/q³，每項非負；總和0恰每個正權重槽Y等於平均，也就是正權重support常值。零權重可異值，非n−1樣本變異數。",
        "en": "E[Y]=M/q and Var=Σ(w/q)(Y−M/q)²=C/q³.Every term is nonnegative,so the sum is0 exactly when EVERY positive-weight Y equals the mean,equivalently constancy on positive support.Zero-weight values may differ.This is NOT n−1 sample variance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。",
          "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "source roles = [scale_variance, 2, null]",
      "after": "derived = [2880/49]",
      "relation": "implies",
      "rule_id": "Variance_SHIFT_SCALE_law_with_conditions",
      "conditions": [],
      "explanation": {
        "zh-Hant": "平移時平均也加c，中心差不變；縮放時平均乘k，中心差乘k故變異數乘k²（k負或0皆可）。此處已以實際Y全量重算，不用定律遮掉未驗c/k。",
        "en": "Under a shift,the mean also shifts by c,leaving deviations unchanged.Under scaling,the mean scales by k and squared deviations by k²,including negative/zero k.The ACTUAL Y values were fully recomputed;the law does not excuse untyped c/k."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。",
          "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "2880/49"
  },
  "conclusion": {
    "canonical_value": "2880/49",
    "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-20261004-advance-20261005-r07-variance-foundations",
      "batch_path": "public/data/questions/batches/batch-20261004-advance-20261005-r07-variance-foundations/questions.jsonl",
      "batch_sha256": "909c76d450ac69e0f6693d74cb693dd27665dd38c1f956808d6f78cb5704e375",
      "original_line": 57,
      "original_line_sha256": "851f69886202db67d7529615262faa0cf48a68dc98595f4a2ec8335ef08c700c",
      "question_compact_sha256": "ca242f0cdd9e5a183eeafeb68d0701593508f836b7bd4ffdc7faa15b714b286b",
      "english_row_compact_sha256": "5ea5ae344014b918ae8467488ccf56a13954c0eeddaee9d14c5c487fec5cdd3d",
      "english_manifest_path": "public/data/i18n/editions/batch-20261004-advance-20261005-r07-variance-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261004-advance-20261005-r07-variance-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-variance": {
        "sha256": "282681c1ed2d55f1c853c9759022470c991dd46c2b150626fa4bf654642032bc",
        "english_sha256": "a07a03bffcd0bef45c5514c2b4ba2db48202bf94f336f0d8589082871649306d",
        "original_path": "public/data/mko/variance.json",
        "english_path": "public/data/i18n/editions/mko-variance.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "20c37a7e64ee5969e7606c5cc3a00e575002ebebdfe82873521cc3d7c608caff",
    "original_candidate_sha256": "2c0d4451f3d8ff0a50896e4d96efb447b0c7bdce6e1ca5d03c52b7a2f68cebec",
    "original_candidate_line_sha256": "2c0d4451f3d8ff0a50896e4d96efb447b0c7bdce6e1ca5d03c52b7a2f68cebec"
  },
  "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-20261026-012-p71",
    "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": "9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80",
      "question_first_witness_sha256": "c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 39,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "4fa7af7f256bac52e98ef395d08d53cfaea5e43d73c6cc34b72ccbda5dcf6d95",
      "arithmetic_steps": 31
    },
    "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."
  }
}