{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261028-r06-random-variable-independence-0083",
  "pilot_number": "R0027-14083",
  "number": 14083,
  "family_id": "qf-mod28-rvind-zero-covariance",
  "title": {
    "zh-Hant": "第 14083 題：隨機變數獨立",
    "en": "Question 14083: Keep Verified Independence and Zero Covariance Distinct"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261028-r06-random-variable-independence-0083",
    "batch_id": "batch-20261014-advance-20261028-r06-random-variable-independence-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-random-variable-independence"
    ],
    "title_zh": "隨機變數獨立",
    "stem_zh": "全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[-5,20],\"y_values\":[-3,27],\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(是, 40/7, 39/7, 1560/49, 0, 是, zero_covariance_converse_not_claimed)",
      "explanation_zh": "Wave4第二層核對=(是, 40/7, 39/7, 1560/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "隨機變數獨立"
      ]
    },
    "variant": {
      "family_id": "qf-mod28-rvind-zero-covariance",
      "seed": 202610286083,
      "parameters": {
        "x_values": [
          -5,
          20
        ],
        "y_values": [
          -3,
          27
        ],
        "masses": [
          [
            [
              20,
              49
            ],
            [
              8,
              49
            ]
          ],
          [
            [
              15,
              49
            ],
            [
              6,
              49
            ]
          ]
        ],
        "domain": "Exact_finite_joint_random_variable_independence",
        "operation": "rvind-zero-covariance",
        "semantic_template": "qf-mod28-rvind-zero-covariance/{\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]],\"x_values\":[-5,20],\"y_values\":[-3,27]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式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-20261028-r06-random-variable-independence-0083",
    "batch_id": "batch-20261014-advance-20261028-r06-random-variable-independence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "d747549ff2544b3f8ff85e4f043d6648aba937fe6069513283725f1673e6fc75",
      "family_id": "qf-mod28-rvind-zero-covariance",
      "seed": 202610286083,
      "semantic_template": "qf-mod28-rvind-zero-covariance/{\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]],\"x_values\":[-5,20],\"y_values\":[-3,27]}"
    },
    "fields": {
      "title": "Keep Verified Independence and Zero Covariance Distinct",
      "stem": "Use the complete joint grid with original x_values=[-5, 20], y_values=[-3, 27], and aligned row-by-column exact masses=[[20/49, 8/49], [15/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.",
      "answer": {
        "kind": "expression",
        "value": "(Yes, 40/7, 39/7, 1560/49, 0, Yes, zero_covariance_converse_not_claimed)",
        "explanation": "Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 40/7, 39/7, 1560/49, 0, Yes, zero_covariance_converse_not_claimed)."
      },
      "skills": [
        "Keep Verified Independence and Zero Covariance Distinct"
      ],
      "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": "全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[-5,20],\"y_values\":[-3,27],\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。",
        "en": "Use the complete joint grid with original x_values=[-5, 20], y_values=[-3, 27], and aligned row-by-column exact masses=[[20/49, 8/49], [15/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "joint grid model whole mass sum := sum(20/49, 8/49, 15/49, 6/49)",
      "after": "joint grid model whole mass sum = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 joint grid model whole mass sum，不以小數近似替代。",
          "en": "Recompute joint grid model whole mass sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20/49",
          "8/49",
          "15/49",
          "6/49"
        ],
        "output": "1",
        "label": "joint grid model whole mass sum"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "joint grid model:raw aligned masses=[[20, 49], [8, 49], [15, 49], [6, 49]]",
      "after": "normalized masses=[20/49, 8/49, 15/49, 6/49];nonnegative=是;total=1",
      "relation": "implies",
      "rule_id": "whole_finite_probability_law_not_uniform_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整同一有限機率模型先核非負與總和恰為1；零質量原子仍保留，不默認均勻或另換模型。",
        "en": "First qualify nonnegativity and exact unit total on one complete finite probability model.Retain zero-mass atoms;do not assume uniformity or substitute another 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
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "original x row labels=[-5, 20];original y column labels=[-3, 27]",
      "after": "normalized aligned row-by-column joint grid=[[20/49, 8/49], [15/49, 6/49]]",
      "relation": "implies",
      "rule_id": "RV_joint_grid_original_order_not_value_sort",
      "conditions": [],
      "explanation": {
        "zh-Hant": "值清單各自不重複，但可非數值排序。所有質量依原行列位置配對，包含零格，不按值排序後錯配矩陣。",
        "en": "Each value list is distinct but need not be numerically sorted.Pair every mass with its original row/column,including zero cells;sorting values must not misalign the grid."
      },
      "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": "X row0 marginal := sum(20/49, 8/49)",
      "after": "X row0 marginal = 4/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 X row0 marginal，不以小數近似替代。",
          "en": "Recompute X row0 marginal with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20/49",
          "8/49"
        ],
        "output": "4/7",
        "label": "X row0 marginal"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "X row1 marginal := sum(15/49, 6/49)",
      "after": "X row1 marginal = 3/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 X row1 marginal，不以小數近似替代。",
          "en": "Recompute X row1 marginal with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "15/49",
          "6/49"
        ],
        "output": "3/7",
        "label": "X row1 marginal"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Y column0 marginal := sum(20/49, 15/49)",
      "after": "Y column0 marginal = 5/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Y column0 marginal，不以小數近似替代。",
          "en": "Recompute Y column0 marginal with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20/49",
          "15/49"
        ],
        "output": "5/7",
        "label": "Y column0 marginal"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Y column1 marginal := sum(8/49, 6/49)",
      "after": "Y column1 marginal = 2/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Y column1 marginal，不以小數近似替代。",
          "en": "Recompute Y column1 marginal with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "8/49",
          "6/49"
        ],
        "output": "2/7",
        "label": "Y column1 marginal"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "joint cell row0 column0 marginal product := mul(4/7, 5/7)",
      "after": "joint cell row0 column0 marginal product = 20/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 joint cell row0 column0 marginal product，不以小數近似替代。",
          "en": "Recompute joint cell row0 column0 marginal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4/7",
          "5/7"
        ],
        "output": "20/49",
        "label": "joint cell row0 column0 marginal product"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "original row=0;column=0;value labels=[-5, -3];joint mass=20/49;marginal product=20/49",
      "after": "cell factorizes=是",
      "relation": "implies",
      "rule_id": "RV_independence_every_joint_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整grid每格核單點事件乘積，零格也不能略過；一個失敗格就給單點逆像事件反例，失敗清單維持原row-major次序。",
        "en": "Audit every cell's singleton-event product,including zeros.One failed cell gives a singleton-preimage counterexample.Retain original row-major failure order."
      },
      "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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "joint cell row0 column1 marginal product := mul(4/7, 2/7)",
      "after": "joint cell row0 column1 marginal product = 8/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 joint cell row0 column1 marginal product，不以小數近似替代。",
          "en": "Recompute joint cell row0 column1 marginal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4/7",
          "2/7"
        ],
        "output": "8/49",
        "label": "joint cell row0 column1 marginal product"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "original row=0;column=1;value labels=[-5, 27];joint mass=8/49;marginal product=8/49",
      "after": "cell factorizes=是",
      "relation": "implies",
      "rule_id": "RV_independence_every_joint_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整grid每格核單點事件乘積，零格也不能略過；一個失敗格就給單點逆像事件反例，失敗清單維持原row-major次序。",
        "en": "Audit every cell's singleton-event product,including zeros.One failed cell gives a singleton-preimage counterexample.Retain original row-major failure order."
      },
      "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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "joint cell row1 column0 marginal product := mul(3/7, 5/7)",
      "after": "joint cell row1 column0 marginal product = 15/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 joint cell row1 column0 marginal product，不以小數近似替代。",
          "en": "Recompute joint cell row1 column0 marginal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3/7",
          "5/7"
        ],
        "output": "15/49",
        "label": "joint cell row1 column0 marginal product"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "original row=1;column=0;value labels=[20, -3];joint mass=15/49;marginal product=15/49",
      "after": "cell factorizes=是",
      "relation": "implies",
      "rule_id": "RV_independence_every_joint_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整grid每格核單點事件乘積，零格也不能略過；一個失敗格就給單點逆像事件反例，失敗清單維持原row-major次序。",
        "en": "Audit every cell's singleton-event product,including zeros.One failed cell gives a singleton-preimage counterexample.Retain original row-major failure order."
      },
      "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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "joint cell row1 column1 marginal product := mul(3/7, 2/7)",
      "after": "joint cell row1 column1 marginal product = 6/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 joint cell row1 column1 marginal product，不以小數近似替代。",
          "en": "Recompute joint cell row1 column1 marginal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3/7",
          "2/7"
        ],
        "output": "6/49",
        "label": "joint cell row1 column1 marginal product"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "original row=1;column=1;value labels=[20, 27];joint mass=6/49;marginal product=6/49",
      "after": "cell factorizes=是",
      "relation": "implies",
      "rule_id": "RV_independence_every_joint_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整grid每格核單點事件乘積，零格也不能略過；一個失敗格就給單點逆像事件反例，失敗清單維持原row-major次序。",
        "en": "Audit every cell's singleton-event product,including zeros.One failed cell gives a singleton-preimage counterexample.Retain original row-major failure order."
      },
      "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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "all joint factorization failures=[]",
      "after": "random variables independent=是;all-cell factorization covers every value-event rectangle",
      "relation": "implies",
      "rule_id": "RV_all_cells_to_all_event_preimages",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全部格分解時，任意值事件矩形的聯合和可分成兩個邊際和的乘積，所以覆蓋所有逆像事件；不推出同分布、IID、Markov或更大家族互獨立。",
        "en": "If every cell factorizes,any value-event rectangle sum splits into a product of marginal sums,covering every preimage-event choice.This implies no identical distribution,IID,Markov behavior or larger-family mutual independence."
      },
      "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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "moment row0 col0 weighted X := mul(20/49, -5)",
      "after": "moment row0 col0 weighted X = -100/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row0 col0 weighted X，不以小數近似替代。",
          "en": "Recompute moment row0 col0 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20/49",
          "-5"
        ],
        "output": "-100/49",
        "label": "moment row0 col0 weighted X"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "moment row0 col0 weighted Y := mul(20/49, -3)",
      "after": "moment row0 col0 weighted Y = -60/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row0 col0 weighted Y，不以小數近似替代。",
          "en": "Recompute moment row0 col0 weighted Y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20/49",
          "-3"
        ],
        "output": "-60/49",
        "label": "moment row0 col0 weighted Y"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "moment row0 col0 XY := mul(-5, -3)",
      "after": "moment row0 col0 XY = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row0 col0 XY，不以小數近似替代。",
          "en": "Recompute moment row0 col0 XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "-3"
        ],
        "output": "15",
        "label": "moment row0 col0 XY"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "moment row0 col0 weighted XY := mul(20/49, 15)",
      "after": "moment row0 col0 weighted XY = 300/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row0 col0 weighted XY，不以小數近似替代。",
          "en": "Recompute moment row0 col0 weighted XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20/49",
          "15"
        ],
        "output": "300/49",
        "label": "moment row0 col0 weighted XY"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "moment row0 col1 weighted X := mul(8/49, -5)",
      "after": "moment row0 col1 weighted X = -40/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row0 col1 weighted X，不以小數近似替代。",
          "en": "Recompute moment row0 col1 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8/49",
          "-5"
        ],
        "output": "-40/49",
        "label": "moment row0 col1 weighted X"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "moment row0 col1 weighted Y := mul(8/49, 27)",
      "after": "moment row0 col1 weighted Y = 216/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row0 col1 weighted Y，不以小數近似替代。",
          "en": "Recompute moment row0 col1 weighted Y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8/49",
          "27"
        ],
        "output": "216/49",
        "label": "moment row0 col1 weighted Y"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "moment row0 col1 XY := mul(-5, 27)",
      "after": "moment row0 col1 XY = -135",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row0 col1 XY，不以小數近似替代。",
          "en": "Recompute moment row0 col1 XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "27"
        ],
        "output": "-135",
        "label": "moment row0 col1 XY"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "moment row0 col1 weighted XY := mul(8/49, -135)",
      "after": "moment row0 col1 weighted XY = -1080/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row0 col1 weighted XY，不以小數近似替代。",
          "en": "Recompute moment row0 col1 weighted XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8/49",
          "-135"
        ],
        "output": "-1080/49",
        "label": "moment row0 col1 weighted XY"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "moment row1 col0 weighted X := mul(15/49, 20)",
      "after": "moment row1 col0 weighted X = 300/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row1 col0 weighted X，不以小數近似替代。",
          "en": "Recompute moment row1 col0 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15/49",
          "20"
        ],
        "output": "300/49",
        "label": "moment row1 col0 weighted X"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "moment row1 col0 weighted Y := mul(15/49, -3)",
      "after": "moment row1 col0 weighted Y = -45/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row1 col0 weighted Y，不以小數近似替代。",
          "en": "Recompute moment row1 col0 weighted Y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15/49",
          "-3"
        ],
        "output": "-45/49",
        "label": "moment row1 col0 weighted Y"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "moment row1 col0 XY := mul(20, -3)",
      "after": "moment row1 col0 XY = -60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row1 col0 XY，不以小數近似替代。",
          "en": "Recompute moment row1 col0 XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "-3"
        ],
        "output": "-60",
        "label": "moment row1 col0 XY"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "moment row1 col0 weighted XY := mul(15/49, -60)",
      "after": "moment row1 col0 weighted XY = -900/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row1 col0 weighted XY，不以小數近似替代。",
          "en": "Recompute moment row1 col0 weighted XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15/49",
          "-60"
        ],
        "output": "-900/49",
        "label": "moment row1 col0 weighted XY"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "moment row1 col1 weighted X := mul(6/49, 20)",
      "after": "moment row1 col1 weighted X = 120/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row1 col1 weighted X，不以小數近似替代。",
          "en": "Recompute moment row1 col1 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6/49",
          "20"
        ],
        "output": "120/49",
        "label": "moment row1 col1 weighted X"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "moment row1 col1 weighted Y := mul(6/49, 27)",
      "after": "moment row1 col1 weighted Y = 162/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row1 col1 weighted Y，不以小數近似替代。",
          "en": "Recompute moment row1 col1 weighted Y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6/49",
          "27"
        ],
        "output": "162/49",
        "label": "moment row1 col1 weighted Y"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "moment row1 col1 XY := mul(20, 27)",
      "after": "moment row1 col1 XY = 540",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row1 col1 XY，不以小數近似替代。",
          "en": "Recompute moment row1 col1 XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "27"
        ],
        "output": "540",
        "label": "moment row1 col1 XY"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "moment row1 col1 weighted XY := mul(6/49, 540)",
      "after": "moment row1 col1 weighted XY = 3240/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 moment row1 col1 weighted XY，不以小數近似替代。",
          "en": "Recompute moment row1 col1 weighted XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6/49",
          "540"
        ],
        "output": "3240/49",
        "label": "moment row1 col1 weighted XY"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "grid E_X := sum(-100/49, -40/49, 300/49, 120/49)",
      "after": "grid E_X = 40/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 grid E_X，不以小數近似替代。",
          "en": "Recompute grid E_X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-100/49",
          "-40/49",
          "300/49",
          "120/49"
        ],
        "output": "40/7",
        "label": "grid E_X"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "grid E_Y := sum(-60/49, 216/49, -45/49, 162/49)",
      "after": "grid E_Y = 39/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 grid E_Y，不以小數近似替代。",
          "en": "Recompute grid E_Y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-60/49",
          "216/49",
          "-45/49",
          "162/49"
        ],
        "output": "39/7",
        "label": "grid E_Y"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "grid E_XY := sum(300/49, -1080/49, -900/49, 3240/49)",
      "after": "grid E_XY = 1560/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 grid E_XY，不以小數近似替代。",
          "en": "Recompute grid E_XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "300/49",
          "-1080/49",
          "-900/49",
          "3240/49"
        ],
        "output": "1560/49",
        "label": "grid E_XY"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "grid product of means := mul(40/7, 39/7)",
      "after": "grid product of means = 1560/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 grid product of means，不以小數近似替代。",
          "en": "Recompute grid product of means with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "40/7",
          "39/7"
        ],
        "output": "1560/49",
        "label": "grid product of means"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "grid covariance := sub(1560/49, 1560/49)",
      "after": "grid covariance = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 grid covariance，不以小數近似替代。",
          "en": "Recompute grid covariance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1560/49",
          "1560/49"
        ],
        "output": "0",
        "label": "grid covariance"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "verified independence=是;E_X=40/7;E_Y=39/7;E_XY=1560/49;covariance=0",
      "after": "source implication_pass is conjunction(independent,covariance0)=是,not material implication",
      "relation": "implies",
      "rule_id": "RV_zero_covariance_conjunction_not_converse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若全格獨立已證，有限雙重和分配得E[XY]=E[X]E[Y]，故Cov=0。原欄名雖叫implication_pass，協定其實要兩者同真；相依模型即使Cov=0仍回No，不是否定獨立推出零協方差定理，也不倒推獨立。",
        "en": "Verified all-cell independence distributes the finite double sum to E[XY]=E[X]E[Y],hence Cov=0.Despite the field name implication_pass,the protocol asks both facts to hold.A dependent zero-covariance model still returns No;this neither refutes the theorem nor proves the converse."
      },
      "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": [
      true,
      "40/7",
      "39/7",
      "1560/49",
      "0",
      true,
      "zero_covariance_converse_not_claimed"
    ]
  },
  "conclusion": {
    "canonical_value": "(是, 40/7, 39/7, 1560/49, 0, 是, zero_covariance_converse_not_claimed)",
    "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-20261028-r06-random-variable-independence-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261028-r06-random-variable-independence-foundations/questions.jsonl",
      "batch_sha256": "d01de33201486c30f18c651effaac5b2d50d64207ccccd8f6c9125ce3a4a6c02",
      "original_line": 83,
      "original_line_sha256": "f004cbbd0a60868a7c0f36387a74408f578f51abe3ea82ac8922b0b1255f8ce7",
      "question_compact_sha256": "d747549ff2544b3f8ff85e4f043d6648aba937fe6069513283725f1673e6fc75",
      "english_row_compact_sha256": "7c2174584a419bf9fcec7054c185afc4e4b23df40c53958a1cc82690d7f09ac0",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261028-r06-random-variable-independence-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261028-r06-random-variable-independence-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-random-variable-independence": {
        "sha256": "e8f6e97a13cd93c4cf37384ba6ab187d3ad4ca22c8f56260ed3707df0b8cc94e",
        "english_sha256": "d30f20ac634c6f78b55166d2cdea3b6fa5631c304d8ecd617b39b0a5aae747e8",
        "original_path": "public/data/mko/random-variable-independence.json",
        "english_path": "public/data/i18n/editions/mko-random-variable-independence.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "a8b6a7c538e8e95c336dfae507795cf07012b597b3dd6104fa3ecad599076f77",
    "original_candidate_sha256": "46808684985fa11d9b0ccf1a5d211221775b9b017e2740f9b44ec0882aaee11e",
    "original_candidate_line_sha256": "46808684985fa11d9b0ccf1a5d211221775b9b017e2740f9b44ec0882aaee11e"
  },
  "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-20261026-007-p28",
    "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": "e49bf5eaa968c09ebac4ad538ce58251671ed922c957cee1471cfc7b68934bfe",
      "question_first_witness_sha256": "a7976057c26051fab92342fc95d48af224466c0e1e4ab6ae4c8811c02772e6da"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 38,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "8105e78f6cc52a06168153cd1d614f89b3b73eff4f164b9ab6a53f2d2ec36c90",
      "arithmetic_steps": 30
    },
    "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."
  }
}