{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261106-r04-estimator-variance-0030",
  "pilot_number": "R0038-19730",
  "number": 19730,
  "family_id": "qf-mod37-estvar-family-readback",
  "title": {
    "zh-Hant": "第 19730 題：估計量變異數",
    "en": "Question 19730: Estimator Variance Remains Parameter-Indexed"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261106-r04-estimator-variance-0030",
    "batch_id": "batch-20261014-advance-20261106-r04-estimator-variance-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-estimator-variance"
    ],
    "title_zh": "估計量變異數",
    "stem_zh": "共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[0, 38/7, 313/49], [1, 5, 23/3]], variance_is_parameter_indexed_family)",
      "explanation_zh": "Wave5封閉層核對=([[0, 38/7, 313/49], [1, 5, 23/3]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "估計量變異數"
      ]
    },
    "variant": {
      "family_id": "qf-mod37-estvar-family-readback",
      "seed": 202611064030,
      "parameters": {
        "laws": [
          [
            [
              1,
              14
            ],
            [
              2,
              14
            ],
            [
              3,
              14
            ],
            [
              8,
              14
            ]
          ],
          [
            [
              2,
              18
            ],
            [
              3,
              18
            ],
            [
              4,
              18
            ],
            [
              9,
              18
            ]
          ]
        ],
        "statistic": [
          0,
          1,
          6,
          7
        ],
        "domain": "Finite_parametric_estimator_variance",
        "operation": "estvar-family-readback",
        "semantic_template": "qf-mod37-estvar-family-readback/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261106-r04-estimator-variance-0030",
    "batch_id": "batch-20261014-advance-20261106-r04-estimator-variance-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "a02aabfbf8fc94668266dc35d5c9207c96f943f01c2b7f3068d5495b131796c3",
      "family_id": "qf-mod37-estvar-family-readback",
      "seed": 202611064030,
      "semantic_template": "qf-mod37-estvar-family-readback/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7]}"
    },
    "fields": {
      "title": "Estimator Variance Remains Parameter-Indexed",
      "stem": "Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([[0, 38/7, 313/49], [1, 5, 23/3]], variance_is_parameter_indexed_family)",
        "explanation": "For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 38/7, 313/49], [1, 5, 23/3]], variance_is_parameter_indexed_family)."
      },
      "skills": [
        "Estimator Variance Remains Parameter-Indexed"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=estvar-family-readback;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[parameter_variance_rows, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。欄位標籤不自動成為數學證據，No依各題聲稱範圍解讀。",
        "en": "Keep the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not mathematical evidence,and No retains the operation's claim scope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "variance parameter0 complete law mass := sum(1/14, 1/7, 3/14, 4/7)",
      "after": "variance parameter0 complete law mass = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 complete law mass，不以小數近似替代。",
          "en": "Recompute variance parameter0 complete law mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/14",
          "1/7",
          "3/14",
          "4/7"
        ],
        "output": "1",
        "label": "variance parameter0 complete law mass"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "original parameter SLOT=0;ALL original outcome masses=[1/14, 1/7, 3/14, 4/7];ONE fixed statistic=[0, 1, 6, 7]",
      "after": "complete same-law mass=1;all masses nonnegative;positive-mass support=[0, 1, 2, 3];null outcomes RETAINED",
      "relation": "implies",
      "rule_id": "estvar_full_law_statistic_null_support_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "各參數law各自合格且總和1；同一T定義不依未知參數，但law、mean與variance依各row計。null結果仍在原表，只是不在該law的正質量support。",
        "en": "Qualify each complete parameter law separately with total1.The same statistic definition is parameter-independent,but laws,means and variances remain row-specific.Keep null outcomes in the original table even though they are outside that law's positive-mass support."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "variance parameter1 complete law mass := sum(1/9, 1/6, 2/9, 1/2)",
      "after": "variance parameter1 complete law mass = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 complete law mass，不以小數近似替代。",
          "en": "Recompute variance parameter1 complete law mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/9",
          "1/6",
          "2/9",
          "1/2"
        ],
        "output": "1",
        "label": "variance parameter1 complete law mass"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "original parameter SLOT=1;ALL original outcome masses=[1/9, 1/6, 2/9, 1/2];ONE fixed statistic=[0, 1, 6, 7]",
      "after": "complete same-law mass=1;all masses nonnegative;positive-mass support=[0, 1, 2, 3];null outcomes RETAINED",
      "relation": "implies",
      "rule_id": "estvar_full_law_statistic_null_support_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "各參數law各自合格且總和1；同一T定義不依未知參數，但law、mean與variance依各row計。null結果仍在原表，只是不在該law的正質量support。",
        "en": "Qualify each complete parameter law separately with total1.The same statistic definition is parameter-independent,but laws,means and variances remain row-specific.Keep null outcomes in the original table even though they are outside that law's positive-mass support."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "variance parameter0 outcome0 weighted T := mul(1/14, 0)",
      "after": "variance parameter0 outcome0 weighted T = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome0 weighted T，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome0 weighted T with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/14",
          "0"
        ],
        "output": "0",
        "label": "variance parameter0 outcome0 weighted T"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "variance parameter0 outcome1 weighted T := mul(1/7, 1)",
      "after": "variance parameter0 outcome1 weighted T = 1/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome1 weighted T，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome1 weighted T with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/7",
          "1"
        ],
        "output": "1/7",
        "label": "variance parameter0 outcome1 weighted T"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "variance parameter0 outcome2 weighted T := mul(3/14, 6)",
      "after": "variance parameter0 outcome2 weighted T = 9/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome2 weighted T，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome2 weighted T with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3/14",
          "6"
        ],
        "output": "9/7",
        "label": "variance parameter0 outcome2 weighted T"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "variance parameter0 outcome3 weighted T := mul(4/7, 7)",
      "after": "variance parameter0 outcome3 weighted T = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome3 weighted T，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome3 weighted T with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4/7",
          "7"
        ],
        "output": "4",
        "label": "variance parameter0 outcome3 weighted T"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "variance parameter0 SAME-law mean := sum(0, 1/7, 9/7, 4)",
      "after": "variance parameter0 SAME-law mean = 38/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 SAME-law mean，不以小數近似替代。",
          "en": "Recompute variance parameter0 SAME-law mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/7",
          "9/7",
          "4"
        ],
        "output": "38/7",
        "label": "variance parameter0 SAME-law mean"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "variance parameter0 outcome0 T minus SAME mean := sub(0, 38/7)",
      "after": "variance parameter0 outcome0 T minus SAME mean = -38/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome0 T minus SAME mean，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome0 T minus SAME mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "38/7"
        ],
        "output": "-38/7",
        "label": "variance parameter0 outcome0 T minus SAME mean"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "variance parameter0 outcome0 centered square := mul(-38/7, -38/7)",
      "after": "variance parameter0 outcome0 centered square = 1444/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome0 centered square，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome0 centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-38/7",
          "-38/7"
        ],
        "output": "1444/49",
        "label": "variance parameter0 outcome0 centered square"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "variance parameter0 outcome0 weighted centered square := mul(1/14, 1444/49)",
      "after": "variance parameter0 outcome0 weighted centered square = 722/343",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome0 weighted centered square，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome0 weighted centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/14",
          "1444/49"
        ],
        "output": "722/343",
        "label": "variance parameter0 outcome0 weighted centered square"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "variance parameter0 outcome1 T minus SAME mean := sub(1, 38/7)",
      "after": "variance parameter0 outcome1 T minus SAME mean = -31/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome1 T minus SAME mean，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome1 T minus SAME mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "38/7"
        ],
        "output": "-31/7",
        "label": "variance parameter0 outcome1 T minus SAME mean"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "variance parameter0 outcome1 centered square := mul(-31/7, -31/7)",
      "after": "variance parameter0 outcome1 centered square = 961/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome1 centered square，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome1 centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-31/7",
          "-31/7"
        ],
        "output": "961/49",
        "label": "variance parameter0 outcome1 centered square"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "variance parameter0 outcome1 weighted centered square := mul(1/7, 961/49)",
      "after": "variance parameter0 outcome1 weighted centered square = 961/343",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome1 weighted centered square，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome1 weighted centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/7",
          "961/49"
        ],
        "output": "961/343",
        "label": "variance parameter0 outcome1 weighted centered square"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "variance parameter0 outcome2 T minus SAME mean := sub(6, 38/7)",
      "after": "variance parameter0 outcome2 T minus SAME mean = 4/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome2 T minus SAME mean，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome2 T minus SAME mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "6",
          "38/7"
        ],
        "output": "4/7",
        "label": "variance parameter0 outcome2 T minus SAME mean"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "variance parameter0 outcome2 centered square := mul(4/7, 4/7)",
      "after": "variance parameter0 outcome2 centered square = 16/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome2 centered square，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome2 centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4/7",
          "4/7"
        ],
        "output": "16/49",
        "label": "variance parameter0 outcome2 centered square"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "variance parameter0 outcome2 weighted centered square := mul(3/14, 16/49)",
      "after": "variance parameter0 outcome2 weighted centered square = 24/343",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome2 weighted centered square，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome2 weighted centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3/14",
          "16/49"
        ],
        "output": "24/343",
        "label": "variance parameter0 outcome2 weighted centered square"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "variance parameter0 outcome3 T minus SAME mean := sub(7, 38/7)",
      "after": "variance parameter0 outcome3 T minus SAME mean = 11/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome3 T minus SAME mean，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome3 T minus SAME mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "7",
          "38/7"
        ],
        "output": "11/7",
        "label": "variance parameter0 outcome3 T minus SAME mean"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "variance parameter0 outcome3 centered square := mul(11/7, 11/7)",
      "after": "variance parameter0 outcome3 centered square = 121/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome3 centered square，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome3 centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11/7",
          "11/7"
        ],
        "output": "121/49",
        "label": "variance parameter0 outcome3 centered square"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "variance parameter0 outcome3 weighted centered square := mul(4/7, 121/49)",
      "after": "variance parameter0 outcome3 weighted centered square = 484/343",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 outcome3 weighted centered square，不以小數近似替代。",
          "en": "Recompute variance parameter0 outcome3 weighted centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4/7",
          "121/49"
        ],
        "output": "484/343",
        "label": "variance parameter0 outcome3 weighted centered square"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "variance parameter0 full centered variance := sum(722/343, 961/343, 24/343, 484/343)",
      "after": "variance parameter0 full centered variance = 313/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter0 full centered variance，不以小數近似替代。",
          "en": "Recompute variance parameter0 full centered variance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "722/343",
          "961/343",
          "24/343",
          "484/343"
        ],
        "output": "313/49",
        "label": "variance parameter0 full centered variance"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "FIXED parameter SLOT=0;original same law=[1/14, 1/7, 3/14, 4/7];same statistic=[0, 1, 6, 7];its own mean=38/7;ALL centered deviations=[-38/7, -31/7, 4/7, 11/7]",
      "after": "ALL nonnegative weighted centered squares=[722/343, 961/343, 24/343, 484/343];variance=313/49;variance0 iff EVERY positive-mass outcome has T=THIS mean",
      "relation": "implies",
      "rule_id": "estvar_zero_iff_same_law_almost_sure_not_null_table_constancy",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限個非負項總和為0當且僅當每項為0；正質量處推出T=此mean，零質量處不限制T。這是同一law下幾乎必然常數，不是全表字面常數、無偏或其他參數下常數。",
        "en": "A finite sum of nonnegative terms is0 iff each term is0.At positive masses this forces T to equal this mean;null outcomes impose no restriction.This is almost-sure constancy under the same law,not literal table constancy,unbiasedness or constancy under another parameter."
      },
      "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": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "variance parameter1 outcome0 weighted T := mul(1/9, 0)",
      "after": "variance parameter1 outcome0 weighted T = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome0 weighted T，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome0 weighted T with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/9",
          "0"
        ],
        "output": "0",
        "label": "variance parameter1 outcome0 weighted T"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "variance parameter1 outcome1 weighted T := mul(1/6, 1)",
      "after": "variance parameter1 outcome1 weighted T = 1/6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome1 weighted T，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome1 weighted T with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/6",
          "1"
        ],
        "output": "1/6",
        "label": "variance parameter1 outcome1 weighted T"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "variance parameter1 outcome2 weighted T := mul(2/9, 6)",
      "after": "variance parameter1 outcome2 weighted T = 4/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome2 weighted T，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome2 weighted T with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2/9",
          "6"
        ],
        "output": "4/3",
        "label": "variance parameter1 outcome2 weighted T"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "variance parameter1 outcome3 weighted T := mul(1/2, 7)",
      "after": "variance parameter1 outcome3 weighted T = 7/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome3 weighted T，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome3 weighted T with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "7"
        ],
        "output": "7/2",
        "label": "variance parameter1 outcome3 weighted T"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "variance parameter1 SAME-law mean := sum(0, 1/6, 4/3, 7/2)",
      "after": "variance parameter1 SAME-law mean = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 SAME-law mean，不以小數近似替代。",
          "en": "Recompute variance parameter1 SAME-law mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/6",
          "4/3",
          "7/2"
        ],
        "output": "5",
        "label": "variance parameter1 SAME-law mean"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "variance parameter1 outcome0 T minus SAME mean := sub(0, 5)",
      "after": "variance parameter1 outcome0 T minus SAME mean = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome0 T minus SAME mean，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome0 T minus SAME mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "5"
        ],
        "output": "-5",
        "label": "variance parameter1 outcome0 T minus SAME mean"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "variance parameter1 outcome0 centered square := mul(-5, -5)",
      "after": "variance parameter1 outcome0 centered square = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome0 centered square，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome0 centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "-5"
        ],
        "output": "25",
        "label": "variance parameter1 outcome0 centered square"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "variance parameter1 outcome0 weighted centered square := mul(1/9, 25)",
      "after": "variance parameter1 outcome0 weighted centered square = 25/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome0 weighted centered square，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome0 weighted centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/9",
          "25"
        ],
        "output": "25/9",
        "label": "variance parameter1 outcome0 weighted centered square"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "variance parameter1 outcome1 T minus SAME mean := sub(1, 5)",
      "after": "variance parameter1 outcome1 T minus SAME mean = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome1 T minus SAME mean，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome1 T minus SAME mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "5"
        ],
        "output": "-4",
        "label": "variance parameter1 outcome1 T minus SAME mean"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "variance parameter1 outcome1 centered square := mul(-4, -4)",
      "after": "variance parameter1 outcome1 centered square = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome1 centered square，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome1 centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-4",
          "-4"
        ],
        "output": "16",
        "label": "variance parameter1 outcome1 centered square"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "variance parameter1 outcome1 weighted centered square := mul(1/6, 16)",
      "after": "variance parameter1 outcome1 weighted centered square = 8/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome1 weighted centered square，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome1 weighted centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/6",
          "16"
        ],
        "output": "8/3",
        "label": "variance parameter1 outcome1 weighted centered square"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "variance parameter1 outcome2 T minus SAME mean := sub(6, 5)",
      "after": "variance parameter1 outcome2 T minus SAME mean = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome2 T minus SAME mean，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome2 T minus SAME mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "6",
          "5"
        ],
        "output": "1",
        "label": "variance parameter1 outcome2 T minus SAME mean"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "variance parameter1 outcome2 centered square := mul(1, 1)",
      "after": "variance parameter1 outcome2 centered square = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome2 centered square，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome2 centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "variance parameter1 outcome2 centered square"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "variance parameter1 outcome2 weighted centered square := mul(2/9, 1)",
      "after": "variance parameter1 outcome2 weighted centered square = 2/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome2 weighted centered square，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome2 weighted centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2/9",
          "1"
        ],
        "output": "2/9",
        "label": "variance parameter1 outcome2 weighted centered square"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "variance parameter1 outcome3 T minus SAME mean := sub(7, 5)",
      "after": "variance parameter1 outcome3 T minus SAME mean = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome3 T minus SAME mean，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome3 T minus SAME mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "7",
          "5"
        ],
        "output": "2",
        "label": "variance parameter1 outcome3 T minus SAME mean"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "variance parameter1 outcome3 centered square := mul(2, 2)",
      "after": "variance parameter1 outcome3 centered square = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome3 centered square，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome3 centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "variance parameter1 outcome3 centered square"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "variance parameter1 outcome3 weighted centered square := mul(1/2, 4)",
      "after": "variance parameter1 outcome3 weighted centered square = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 outcome3 weighted centered square，不以小數近似替代。",
          "en": "Recompute variance parameter1 outcome3 weighted centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "4"
        ],
        "output": "2",
        "label": "variance parameter1 outcome3 weighted centered square"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "variance parameter1 full centered variance := sum(25/9, 8/3, 2/9, 2)",
      "after": "variance parameter1 full centered variance = 23/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 variance parameter1 full centered variance，不以小數近似替代。",
          "en": "Recompute variance parameter1 full centered variance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "25/9",
          "8/3",
          "2/9",
          "2"
        ],
        "output": "23/3",
        "label": "variance parameter1 full centered variance"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "FIXED parameter SLOT=1;original same law=[1/9, 1/6, 2/9, 1/2];same statistic=[0, 1, 6, 7];its own mean=5;ALL centered deviations=[-5, -4, 1, 2]",
      "after": "ALL nonnegative weighted centered squares=[25/9, 8/3, 2/9, 2];variance=23/3;variance0 iff EVERY positive-mass outcome has T=THIS mean",
      "relation": "implies",
      "rule_id": "estvar_zero_iff_same_law_almost_sure_not_null_table_constancy",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限個非負項總和為0當且僅當每項為0；正質量處推出T=此mean，零質量處不限制T。這是同一law下幾乎必然常數，不是全表字面常數、無偏或其他參數下常數。",
        "en": "A finite sum of nonnegative terms is0 iff each term is0.At positive masses this forces T to equal this mean;null outcomes impose no restriction.This is almost-sure constancy under the same law,not literal table constancy,unbiasedness or constancy under another parameter."
      },
      "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": [
      [
        [
          0,
          "38/7",
          "313/49"
        ],
        [
          1,
          "5",
          "23/3"
        ]
      ],
      "variance_is_parameter_indexed_family"
    ]
  },
  "conclusion": {
    "canonical_value": "([[0, 38/7, 313/49], [1, 5, 23/3]], variance_is_parameter_indexed_family)",
    "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-20261014-advance-20261106-r04-estimator-variance-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261106-r04-estimator-variance-foundations/questions.jsonl",
      "batch_sha256": "578cc5872162bdbc808ac27796ae60f77bd4166f2127ae321f0eb6543279d70d",
      "original_line": 30,
      "original_line_sha256": "90c2b06f8252d19cf5fe1ba27314de9883e993a55e8bfbc904e492e7902cbad6",
      "question_compact_sha256": "a02aabfbf8fc94668266dc35d5c9207c96f943f01c2b7f3068d5495b131796c3",
      "english_row_compact_sha256": "f2696208fd262f0f35d0590b79d156429fbda7f90ef94d7d22f3a58c0909ae0b",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261106-r04-estimator-variance-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261106-r04-estimator-variance-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-estimator-variance": {
        "sha256": "8e31f372ae9c9485e8d279c1850f034c30f4a1d657c84cd603be31da9a55e53f",
        "english_sha256": "10812015f30a09fe7768421828193eb4aff15eab016edddf0f99b5f3601d9953",
        "original_path": "public/data/mko/estimator-variance.json",
        "english_path": "public/data/i18n/editions/mko-estimator-variance.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "9ed24319f1e4df0dd58d2a2dfbf2846def03f020cb5ebf70613fab53c9165f11",
    "original_candidate_sha256": "be931ed56b5536adc2ec8e7c189b570e4452f3247ea215670e3353334db40ecc",
    "original_candidate_line_sha256": "be931ed56b5536adc2ec8e7c189b570e4452f3247ea215670e3353334db40ecc"
  },
  "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-20261026-008-p39",
    "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": "d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612",
      "question_first_witness_sha256": "c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 43,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "60a295cb1f918aa0dee58458c21e2648703f4ffadcb300d789c3736cbee3e198",
      "arithmetic_steps": 38
    },
    "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."
  }
}