{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261026-r04-finite-joint-distribution-0046",
  "pilot_number": "R0024-12521",
  "number": 12521,
  "family_id": "qf-mod26-joint-marginals",
  "title": {
    "zh-Hant": "第 12521 題：有限聯合分布",
    "en": "Question 12521: Aggregate Both Marginals from the Same Qualified Joint Law"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261026-r04-finite-joint-distribution-0046",
    "batch_id": "batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-joint-distribution"
    ],
    "title_zh": "有限聯合分布",
    "stem_zh": "共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[8,-1],[9,0],[10,1],[11,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[8, 1/8], [9, 1/4], [10, 1/8], [11, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])",
      "explanation_zh": "結構深度核對=([[8, 1/8], [9, 1/4], [10, 1/8], [11, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；質量非負且精確和為1；不外推連續或無限support。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限聯合分布"
      ]
    },
    "variant": {
      "family_id": "qf-mod26-joint-marginals",
      "seed": 202610264046,
      "parameters": {
        "support": [
          [
            8,
            -1
          ],
          [
            9,
            0
          ],
          [
            10,
            1
          ],
          [
            11,
            2
          ]
        ],
        "masses": [
          [
            1,
            8
          ],
          [
            1,
            4
          ],
          [
            1,
            8
          ],
          [
            1,
            2
          ]
        ],
        "domain": "Exact_finite_joint_rational_law",
        "operation": "joint-marginals",
        "semantic_template": "qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[8,-1],[9,0],[10,1],[11,2]]}"
      }
    },
    "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、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261026-r04-finite-joint-distribution-0046",
    "batch_id": "batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2a017684b3932a5f6f09b96c576771e3f4376a685109d5ce7d6252133c23dde5",
      "family_id": "qf-mod26-joint-marginals",
      "seed": 202610264046,
      "semantic_template": "qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[8,-1],[9,0],[10,1],[11,2]]}"
    },
    "fields": {
      "title": "Aggregate Both Marginals from the Same Qualified Joint Law",
      "stem": "Use the complete list of distinct integer pairs=[[8, -1], [9, 0], [10, 1], [11, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.",
      "answer": {
        "kind": "expression",
        "value": "([[8, 1/8], [9, 1/4], [10, 1/8], [11, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])",
        "explanation": "Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[8, 1/8], [9, 1/4], [10, 1/8], [11, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])."
      },
      "skills": [
        "Aggregate Both Marginals from the Same Qualified Joint Law"
      ],
      "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": "共同有限機率空間上的完整joint support與精確質量。操作=joint-marginals；資料={\"support\":[[8,-1],[9,0],[10,1],[11,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [x_marginal,y_marginal] 欄序回答。",
        "en": "Use the complete list of distinct integer pairs=[[8, -1], [9, 0], [10, 1], [11, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "joint law exact total := sum(1/8, 1/4, 1/8, 1/2)",
      "after": "joint law exact total = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 joint law exact total，不以小數近似替代。",
          "en": "Recompute joint law exact total with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/8",
          "1/4",
          "1/8",
          "1/2"
        ],
        "output": "1",
        "label": "joint law exact total"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "original ordered support=[[8, -1], [9, 0], [10, 1], [11, 2]];raw aligned masses=[[1, 8], [1, 4], [1, 8], [1, 2]]",
      "after": "normalized aligned masses=[1/8, 1/4, 1/8, 1/2];nonnegative=是;total=1;qualified=是",
      "relation": "implies",
      "rule_id": "whole_joint_table_not_marginal_substitution",
      "conditions": [],
      "explanation": {
        "zh-Hant": "共同聯合表原序與質量對齊，support對不重複；零質量合法。非負與精確和1分開核，不能用邊際或均勻分布替代原模型。",
        "en": "Keep the complete joint table's original pair-mass alignment with distinct support pairs.Zero masses are allowed.Check nonnegativity and exact unit total separately;do not substitute marginals or a uniform law."
      },
      "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": "marginal coordinate0 label8 := sum(1/8)",
      "after": "marginal coordinate0 label8 = 1/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marginal coordinate0 label8，不以小數近似替代。",
          "en": "Recompute marginal coordinate0 label8 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/8"
        ],
        "output": "1/8",
        "label": "marginal coordinate0 label8"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "coordinate=0;label=8;original support indices=[0]",
      "after": "aggregated marginal mass=1/8;listed label retained even at mass0",
      "relation": "implies",
      "rule_id": "joint_marginal_all_original_atoms",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按固定座標聚合同值原子，再依數值標籤排序；即使聚合質量0也保留原表出現過的標籤，不把兩邊邊際相乘猜聯合表。",
        "en": "Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "marginal coordinate0 label9 := sum(1/4)",
      "after": "marginal coordinate0 label9 = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marginal coordinate0 label9，不以小數近似替代。",
          "en": "Recompute marginal coordinate0 label9 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4"
        ],
        "output": "1/4",
        "label": "marginal coordinate0 label9"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "coordinate=0;label=9;original support indices=[1]",
      "after": "aggregated marginal mass=1/4;listed label retained even at mass0",
      "relation": "implies",
      "rule_id": "joint_marginal_all_original_atoms",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按固定座標聚合同值原子，再依數值標籤排序；即使聚合質量0也保留原表出現過的標籤，不把兩邊邊際相乘猜聯合表。",
        "en": "Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law."
      },
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "marginal coordinate0 label10 := sum(1/8)",
      "after": "marginal coordinate0 label10 = 1/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marginal coordinate0 label10，不以小數近似替代。",
          "en": "Recompute marginal coordinate0 label10 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/8"
        ],
        "output": "1/8",
        "label": "marginal coordinate0 label10"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "coordinate=0;label=10;original support indices=[2]",
      "after": "aggregated marginal mass=1/8;listed label retained even at mass0",
      "relation": "implies",
      "rule_id": "joint_marginal_all_original_atoms",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按固定座標聚合同值原子，再依數值標籤排序；即使聚合質量0也保留原表出現過的標籤，不把兩邊邊際相乘猜聯合表。",
        "en": "Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law."
      },
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "marginal coordinate0 label11 := sum(1/2)",
      "after": "marginal coordinate0 label11 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marginal coordinate0 label11，不以小數近似替代。",
          "en": "Recompute marginal coordinate0 label11 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2"
        ],
        "output": "1/2",
        "label": "marginal coordinate0 label11"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "coordinate=0;label=11;original support indices=[3]",
      "after": "aggregated marginal mass=1/2;listed label retained even at mass0",
      "relation": "implies",
      "rule_id": "joint_marginal_all_original_atoms",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按固定座標聚合同值原子，再依數值標籤排序；即使聚合質量0也保留原表出現過的標籤，不把兩邊邊際相乘猜聯合表。",
        "en": "Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law."
      },
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "marginal coordinate1 label-1 := sum(1/8)",
      "after": "marginal coordinate1 label-1 = 1/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marginal coordinate1 label-1，不以小數近似替代。",
          "en": "Recompute marginal coordinate1 label-1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/8"
        ],
        "output": "1/8",
        "label": "marginal coordinate1 label-1"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "coordinate=1;label=-1;original support indices=[0]",
      "after": "aggregated marginal mass=1/8;listed label retained even at mass0",
      "relation": "implies",
      "rule_id": "joint_marginal_all_original_atoms",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按固定座標聚合同值原子，再依數值標籤排序；即使聚合質量0也保留原表出現過的標籤，不把兩邊邊際相乘猜聯合表。",
        "en": "Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law."
      },
      "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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "marginal coordinate1 label0 := sum(1/4)",
      "after": "marginal coordinate1 label0 = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marginal coordinate1 label0，不以小數近似替代。",
          "en": "Recompute marginal coordinate1 label0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4"
        ],
        "output": "1/4",
        "label": "marginal coordinate1 label0"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "coordinate=1;label=0;original support indices=[1]",
      "after": "aggregated marginal mass=1/4;listed label retained even at mass0",
      "relation": "implies",
      "rule_id": "joint_marginal_all_original_atoms",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按固定座標聚合同值原子，再依數值標籤排序；即使聚合質量0也保留原表出現過的標籤，不把兩邊邊際相乘猜聯合表。",
        "en": "Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law."
      },
      "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": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "marginal coordinate1 label1 := sum(1/8)",
      "after": "marginal coordinate1 label1 = 1/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marginal coordinate1 label1，不以小數近似替代。",
          "en": "Recompute marginal coordinate1 label1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/8"
        ],
        "output": "1/8",
        "label": "marginal coordinate1 label1"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "coordinate=1;label=1;original support indices=[2]",
      "after": "aggregated marginal mass=1/8;listed label retained even at mass0",
      "relation": "implies",
      "rule_id": "joint_marginal_all_original_atoms",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按固定座標聚合同值原子，再依數值標籤排序；即使聚合質量0也保留原表出現過的標籤，不把兩邊邊際相乘猜聯合表。",
        "en": "Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law."
      },
      "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": "marginal coordinate1 label2 := sum(1/2)",
      "after": "marginal coordinate1 label2 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marginal coordinate1 label2，不以小數近似替代。",
          "en": "Recompute marginal coordinate1 label2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2"
        ],
        "output": "1/2",
        "label": "marginal coordinate1 label2"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "coordinate=1;label=2;original support indices=[3]",
      "after": "aggregated marginal mass=1/2;listed label retained even at mass0",
      "relation": "implies",
      "rule_id": "joint_marginal_all_original_atoms",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按固定座標聚合同值原子，再依數值標籤排序；即使聚合質量0也保留原表出現過的標籤，不把兩邊邊際相乘猜聯合表。",
        "en": "Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law."
      },
      "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": [
      [
        [
          8,
          "1/8"
        ],
        [
          9,
          "1/4"
        ],
        [
          10,
          "1/8"
        ],
        [
          11,
          "1/2"
        ]
      ],
      [
        [
          -1,
          "1/8"
        ],
        [
          0,
          "1/4"
        ],
        [
          1,
          "1/8"
        ],
        [
          2,
          "1/2"
        ]
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([[8, 1/8], [9, 1/4], [10, 1/8], [11, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])",
    "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-20261013-advance-20261026-r04-finite-joint-distribution-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations/questions.jsonl",
      "batch_sha256": "c6fea07f9fb42f4d12a8e43b26e58203263484444a91f77a9e5eda122a0dd1b1",
      "original_line": 46,
      "original_line_sha256": "73c773b31bbf04175c1ccc66a5ff5f083fabeb463c85dad0fb3be5b96825b176",
      "question_compact_sha256": "2a017684b3932a5f6f09b96c576771e3f4376a685109d5ce7d6252133c23dde5",
      "english_row_compact_sha256": "5aa3616d34cd0a8b11eb83dae252ba2958c7bdb2e767b1d09d259cdf408f5357",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-joint-distribution": {
        "sha256": "d04aa2266aeda810e9e7e400220d366a61da23cdc33192e7286eac1d0fed0340",
        "english_sha256": "56edeacfb8cda68555a194d3d171ecf5208fd8129a3c2a5884aacd132751bb06",
        "original_path": "public/data/mko/finite-joint-distribution.json",
        "english_path": "public/data/i18n/editions/mko-finite-joint-distribution.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "17a80c8482f64498e5a88bd4368d0a11b92d45c259784a0a536a29fbb98662d5",
    "original_candidate_sha256": "cf7d132c723717f73ed0f54ce12331c04700869caa75e1d519b07e2710278cc9",
    "original_candidate_line_sha256": "cf7d132c723717f73ed0f54ce12331c04700869caa75e1d519b07e2710278cc9"
  },
  "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-20261026-006-p25",
    "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": "1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35",
      "question_first_witness_sha256": "05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 18,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "aa894f82f5b49b3b2f7d9745072bf22bbdc46747cd4ec7b1a448bfd681df3d86",
      "arithmetic_steps": 9
    },
    "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."
  }
}