{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261010-r05-expectation-0071",
  "pilot_number": "R0007-04196",
  "number": 4196,
  "family_id": "qf-density10-expectation-product-pair",
  "title": {
    "zh-Hant": "第 4196 題：乘積期望與均值積的有序對",
    "en": "Question 4196: Product Expectation versus Product of Means"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261010-r05-expectation-0071",
    "batch_id": "batch-20261008-advance-20261010-r05-expectation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-expectation"
    ],
    "title_zh": "乘積期望與均值積的有序對",
    "stem_zh": "有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[21, 2, 1]、D=24，權重為質量/D；X值=[-20, 2, 21]、Y值=[22, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。",
    "answer": {
      "kind": "expression",
      "value": "(-4601/12, -30415/96)",
      "explanation_zh": "乘積期望與均值積=(-4601/12, -30415/96)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "乘積期望與均值積的有序對"
      ]
    },
    "variant": {
      "family_id": "qf-density10-expectation-product-pair",
      "seed": 202610105071,
      "parameters": {
        "masses": [
          21,
          2,
          1
        ],
        "D": 24,
        "values": [
          -20,
          2,
          21
        ],
        "other": [
          22,
          -1,
          2
        ],
        "operation": "density10_expectation_product_pair",
        "domain": "finite_integer_random_variable",
        "semantic_template": "qf-density10-expectation-product-pair/[[-1,2,\"1/12\"],[2,21,\"1/24\"],[22,-20,\"7/8\"]]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-08",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261008-advance-20261010-r05-expectation-0071",
    "batch_id": "batch-20261008-advance-20261010-r05-expectation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "5e944a106fcc6ed6133ccabe108bc8e5c2e5476fa0875003f9698041147427ed",
      "family_id": "qf-density10-expectation-product-pair",
      "seed": 202610105071,
      "semantic_template": "qf-density10-expectation-product-pair/[[-1,2,\"1/12\"],[2,21,\"1/24\"],[22,-20,\"7/8\"]]"
    },
    "fields": {
      "title": "Product Expectation versus Product of Means",
      "stem": "S={0,…,2}, F=P(S), masses=[21, 2, 1], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-20, 2, 21] in index order and Y=[22, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(-4601/12, -30415/96)",
        "explanation": "Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-4601/12, -30415/96)."
      },
      "skills": [
        "Product Expectation versus Product of Means"
      ],
      "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": "有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[21, 2, 1]、D=24，權重為質量/D；X值=[-20, 2, 21]、Y值=[22, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。",
        "en": "S={0,…,2}, F=P(S), masses=[21, 2, 1], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-20, 2, 21] in index order and Y=[22, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "total mass numerator := sum(21, 2, 1)",
      "after": "total mass numerator = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非負整數質量要加總成D，權重才恰歸一化。",
        "en": "Nonnegative integer masses must sum to D for normalized weights."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 total mass numerator，不以小數近似替代。",
          "en": "Recompute total mass numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "2",
          "1"
        ],
        "output": "24",
        "label": "total mass numerator"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "indexed masses=[21, 2, 1];D=24",
      "after": "probabilities=[7/8, 1/12, 1/24]",
      "relation": "implies",
      "rule_id": "indexed_normalized_expectation_model",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本索引從0起，質量、X與Y值按相同索引對齊；零質量結果仍保留，負隨機變數值合法。",
        "en": "Outcome indices start at0 and align masses with X and Y values.Retain zero-mass outcomes;negative random-variable values are allowed."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "X weighted numerator0 := mul(21, -20)",
      "after": "X weighted numerator0 = -420",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將此索引質量乘同索引值，不能刪掉零質量後錯位。",
        "en": "Multiply the mass by the value at the same index;deleting zero masses must not shift alignment."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 X weighted numerator0，不以小數近似替代。",
          "en": "Recompute X weighted numerator0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "-20"
        ],
        "output": "-420",
        "label": "X weighted numerator0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "X contribution0 := div(-420, 24)",
      "after": "X contribution0 = -35/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除以共同正分母；每個零項也保留其索引位置。",
        "en": "Divide by the common positive denominator,retaining the index position of a zero contribution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 X contribution0，不以小數近似替代。",
          "en": "Recompute X contribution0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-420",
          "24"
        ],
        "output": "-35/2",
        "label": "X contribution0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "X weighted numerator1 := mul(2, 2)",
      "after": "X weighted numerator1 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將此索引質量乘同索引值，不能刪掉零質量後錯位。",
        "en": "Multiply the mass by the value at the same index;deleting zero masses must not shift alignment."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 X weighted numerator1，不以小數近似替代。",
          "en": "Recompute X weighted numerator1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "X weighted numerator1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "X contribution1 := div(4, 24)",
      "after": "X contribution1 = 1/6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除以共同正分母；每個零項也保留其索引位置。",
        "en": "Divide by the common positive denominator,retaining the index position of a zero contribution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 X contribution1，不以小數近似替代。",
          "en": "Recompute X contribution1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "4",
          "24"
        ],
        "output": "1/6",
        "label": "X contribution1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "X weighted numerator2 := mul(1, 21)",
      "after": "X weighted numerator2 = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將此索引質量乘同索引值，不能刪掉零質量後錯位。",
        "en": "Multiply the mass by the value at the same index;deleting zero masses must not shift alignment."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 X weighted numerator2，不以小數近似替代。",
          "en": "Recompute X weighted numerator2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "21"
        ],
        "output": "21",
        "label": "X weighted numerator2"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "X contribution2 := div(21, 24)",
      "after": "X contribution2 = 7/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除以共同正分母；每個零項也保留其索引位置。",
        "en": "Divide by the common positive denominator,retaining the index position of a zero contribution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 X contribution2，不以小數近似替代。",
          "en": "Recompute X contribution2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "21",
          "24"
        ],
        "output": "7/8",
        "label": "X contribution2"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "X expectation := sum(-35/2, 1/6, 7/8)",
      "after": "X expectation = -395/24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整相加全部逐點期望項。",
        "en": "Add every indexed expectation contribution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 X expectation，不以小數近似替代。",
          "en": "Recompute X expectation with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-35/2",
          "1/6",
          "7/8"
        ],
        "output": "-395/24",
        "label": "X expectation"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Y weighted numerator0 := mul(21, 22)",
      "after": "Y weighted numerator0 = 462",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將此索引質量乘同索引值，不能刪掉零質量後錯位。",
        "en": "Multiply the mass by the value at the same index;deleting zero masses must not shift alignment."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Y weighted numerator0，不以小數近似替代。",
          "en": "Recompute Y weighted numerator0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "22"
        ],
        "output": "462",
        "label": "Y weighted numerator0"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Y contribution0 := div(462, 24)",
      "after": "Y contribution0 = 77/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除以共同正分母；每個零項也保留其索引位置。",
        "en": "Divide by the common positive denominator,retaining the index position of a zero contribution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Y contribution0，不以小數近似替代。",
          "en": "Recompute Y contribution0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "24"
        ],
        "output": "77/4",
        "label": "Y contribution0"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Y weighted numerator1 := mul(2, -1)",
      "after": "Y weighted numerator1 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將此索引質量乘同索引值，不能刪掉零質量後錯位。",
        "en": "Multiply the mass by the value at the same index;deleting zero masses must not shift alignment."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Y weighted numerator1，不以小數近似替代。",
          "en": "Recompute Y weighted numerator1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-1"
        ],
        "output": "-2",
        "label": "Y weighted numerator1"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Y contribution1 := div(-2, 24)",
      "after": "Y contribution1 = -1/12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除以共同正分母；每個零項也保留其索引位置。",
        "en": "Divide by the common positive denominator,retaining the index position of a zero contribution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Y contribution1，不以小數近似替代。",
          "en": "Recompute Y contribution1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-2",
          "24"
        ],
        "output": "-1/12",
        "label": "Y contribution1"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Y weighted numerator2 := mul(1, 2)",
      "after": "Y weighted numerator2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將此索引質量乘同索引值，不能刪掉零質量後錯位。",
        "en": "Multiply the mass by the value at the same index;deleting zero masses must not shift alignment."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Y weighted numerator2，不以小數近似替代。",
          "en": "Recompute Y weighted numerator2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "Y weighted numerator2"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Y contribution2 := div(2, 24)",
      "after": "Y contribution2 = 1/12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除以共同正分母；每個零項也保留其索引位置。",
        "en": "Divide by the common positive denominator,retaining the index position of a zero contribution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Y contribution2，不以小數近似替代。",
          "en": "Recompute Y contribution2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "2",
          "24"
        ],
        "output": "1/12",
        "label": "Y contribution2"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Y expectation := sum(77/4, -1/12, 1/12)",
      "after": "Y expectation = 77/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整相加全部逐點期望項。",
        "en": "Add every indexed expectation contribution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Y expectation，不以小數近似替代。",
          "en": "Recompute Y expectation with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "77/4",
          "-1/12",
          "1/12"
        ],
        "output": "77/4",
        "label": "Y expectation"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "pointwise XY0 := mul(-20, 22)",
      "after": "pointwise XY0 = -440",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先在相同基本結果上相乘X_iY_i，再套原權重。",
        "en": "First multiply X_iY_i at the same outcome,then apply its original weight."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 pointwise XY0，不以小數近似替代。",
          "en": "Recompute pointwise XY0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-20",
          "22"
        ],
        "output": "-440",
        "label": "pointwise XY0"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "pointwise XY1 := mul(2, -1)",
      "after": "pointwise XY1 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先在相同基本結果上相乘X_iY_i，再套原權重。",
        "en": "First multiply X_iY_i at the same outcome,then apply its original weight."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 pointwise XY1，不以小數近似替代。",
          "en": "Recompute pointwise XY1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-1"
        ],
        "output": "-2",
        "label": "pointwise XY1"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "pointwise XY2 := mul(21, 2)",
      "after": "pointwise XY2 = 42",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先在相同基本結果上相乘X_iY_i，再套原權重。",
        "en": "First multiply X_iY_i at the same outcome,then apply its original weight."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 pointwise XY2，不以小數近似替代。",
          "en": "Recompute pointwise XY2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "2"
        ],
        "output": "42",
        "label": "pointwise XY2"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "XY weighted numerator0 := mul(21, -440)",
      "after": "XY weighted numerator0 = -9240",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將此索引質量乘同索引值，不能刪掉零質量後錯位。",
        "en": "Multiply the mass by the value at the same index;deleting zero masses must not shift alignment."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 XY weighted numerator0，不以小數近似替代。",
          "en": "Recompute XY weighted numerator0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "-440"
        ],
        "output": "-9240",
        "label": "XY weighted numerator0"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "XY contribution0 := div(-9240, 24)",
      "after": "XY contribution0 = -385",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除以共同正分母；每個零項也保留其索引位置。",
        "en": "Divide by the common positive denominator,retaining the index position of a zero contribution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 XY contribution0，不以小數近似替代。",
          "en": "Recompute XY contribution0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-9240",
          "24"
        ],
        "output": "-385",
        "label": "XY contribution0"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "XY weighted numerator1 := mul(2, -2)",
      "after": "XY weighted numerator1 = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將此索引質量乘同索引值，不能刪掉零質量後錯位。",
        "en": "Multiply the mass by the value at the same index;deleting zero masses must not shift alignment."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 XY weighted numerator1，不以小數近似替代。",
          "en": "Recompute XY weighted numerator1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-2"
        ],
        "output": "-4",
        "label": "XY weighted numerator1"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "XY contribution1 := div(-4, 24)",
      "after": "XY contribution1 = -1/6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除以共同正分母；每個零項也保留其索引位置。",
        "en": "Divide by the common positive denominator,retaining the index position of a zero contribution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 XY contribution1，不以小數近似替代。",
          "en": "Recompute XY contribution1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-4",
          "24"
        ],
        "output": "-1/6",
        "label": "XY contribution1"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "XY weighted numerator2 := mul(1, 42)",
      "after": "XY weighted numerator2 = 42",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將此索引質量乘同索引值，不能刪掉零質量後錯位。",
        "en": "Multiply the mass by the value at the same index;deleting zero masses must not shift alignment."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 XY weighted numerator2，不以小數近似替代。",
          "en": "Recompute XY weighted numerator2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "42"
        ],
        "output": "42",
        "label": "XY weighted numerator2"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "XY contribution2 := div(42, 24)",
      "after": "XY contribution2 = 7/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除以共同正分母；每個零項也保留其索引位置。",
        "en": "Divide by the common positive denominator,retaining the index position of a zero contribution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 XY contribution2，不以小數近似替代。",
          "en": "Recompute XY contribution2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "42",
          "24"
        ],
        "output": "7/4",
        "label": "XY contribution2"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "XY expectation := sum(-385, -1/6, 7/4)",
      "after": "XY expectation = -4601/12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整相加全部逐點期望項。",
        "en": "Add every indexed expectation contribution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 XY expectation，不以小數近似替代。",
          "en": "Recompute XY expectation with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-385",
          "-1/6",
          "7/4"
        ],
        "output": "-4601/12",
        "label": "XY expectation"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "product of means := mul(-395/24, 77/4)",
      "after": "product of means = -30415/96",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "另算兩均值的乘積，不把它直接代作乘積的期望。",
        "en": "Compute the product of means separately;do not substitute it for the expectation of the product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 product of means，不以小數近似替代。",
          "en": "Recompute product of means with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-395/24",
          "77/4"
        ],
        "output": "-30415/96",
        "label": "product of means"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "E[XY]=-4601/12;E[X]E[Y]=-30415/96",
      "after": "report in this order;no independence inferred",
      "relation": "implies",
      "rule_id": "mixed_moment_not_independence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "獨立可推出此矩關係，但一個矩關係相等本身不推出一般獨立；此處只計算明示共模型的兩個數。",
        "en": "Independence implies this moment equality,but one moment equality alone does not establish general independence;compute only the two values in the declared joint model."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "-4601/12",
      "-30415/96"
    ],
    "means": [
      "-395/24",
      "77/4"
    ]
  },
  "conclusion": {
    "canonical_value": "(-4601/12, -30415/96)",
    "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-20261008-advance-20261010-r05-expectation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261010-r05-expectation-foundations/questions.jsonl",
      "batch_sha256": "4ff64bd7b738bff274c56f73708b9491e71e052b39893de82b63de7618bd6fa2",
      "original_line": 71,
      "original_line_sha256": "f53cd17c09756eaff159a2eccb75285c00a06494f59ee898023d1a30e8cb710c",
      "question_compact_sha256": "5e944a106fcc6ed6133ccabe108bc8e5c2e5476fa0875003f9698041147427ed",
      "english_row_compact_sha256": "2753a70699334f6552a3aafb47c16addd1d94126e7517f349532bef1efbfb6af",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261010-r05-expectation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261010-r05-expectation-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-expectation": {
        "sha256": "71bc68faf69ca0df9cdd11d093055262aa277d07abd77982d0aa1afd849daf3c",
        "english_sha256": "636015fa1e3a6d9eae6091042ae3dd70c6cb996ecd9a409c890060a5b503333a",
        "original_path": "public/data/mko/expectation.json",
        "english_path": "public/data/i18n/editions/mko-expectation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "de1fb611cf230f243ddd3334fbc53c9205cbe528b78346a10544972dede78b95",
    "original_candidate_sha256": "fb73d90cc61779036f6da93842897f1703244e5ff7f86324d2d19ceae6bd4126",
    "original_candidate_line_sha256": "fb73d90cc61779036f6da93842897f1703244e5ff7f86324d2d19ceae6bd4126"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-004-p8",
    "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": "faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a",
      "question_first_witness_sha256": "4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 28,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "0b1ea4f0e884528e3d5f01aca197e4a08ef286810a7d5d647f9a587d86758d76",
      "arithmetic_steps": 26
    },
    "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."
  }
}