{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261026-r05-covariance-0036",
  "pilot_number": "R0023-12261",
  "number": 12261,
  "family_id": "qf-mod26-covariance-symmetry",
  "title": {
    "zh-Hant": "第 12261 題：協方差",
    "en": "Question 12261: Symmetry of Covariance under Swapping Coordinates"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261026-r05-covariance-0036",
    "batch_id": "batch-20261013-advance-20261026-r05-covariance-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-covariance"
    ],
    "title_zh": "協方差",
    "stem_zh": "協方差必須綁定同一完整有限joint law。操作=covariance-symmetry；資料={\"support\":[[-2,-1],[-1,0],[0,1],[1,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [Cov_XY,Cov_YX,equal] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(5/4, 5/4, 是)",
      "explanation_zh": "結構深度核對=(5/4, 5/4, 是)；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；非零協方差可否證獨立；零協方差通常不推出獨立。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "協方差"
      ]
    },
    "variant": {
      "family_id": "qf-mod26-covariance-symmetry",
      "seed": 202610265036,
      "parameters": {
        "support": [
          [
            -2,
            -1
          ],
          [
            -1,
            0
          ],
          [
            0,
            1
          ],
          [
            1,
            2
          ]
        ],
        "masses": [
          [
            1,
            8
          ],
          [
            1,
            4
          ],
          [
            1,
            8
          ],
          [
            1,
            2
          ]
        ],
        "domain": "Exact_finite_joint_covariance",
        "operation": "covariance-symmetry",
        "semantic_template": "qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-2,-1],[-1,0],[0,1],[1,2]]}"
      }
    },
    "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-r05-covariance-0036",
    "batch_id": "batch-20261013-advance-20261026-r05-covariance-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "15ffc3b428f006efc6917be6660becf0a03ce04c79ea4d570a671e35ea1de510",
      "family_id": "qf-mod26-covariance-symmetry",
      "seed": 202610265036,
      "semantic_template": "qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-2,-1],[-1,0],[0,1],[1,2]]}"
    },
    "fields": {
      "title": "Symmetry of Covariance under Swapping Coordinates",
      "stem": "Use the complete list of distinct integer pairs=[[-2, -1], [-1, 0], [0, 1], [1, 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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).",
      "answer": {
        "kind": "expression",
        "value": "(5/4, 5/4, Yes)",
        "explanation": "Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/4, 5/4, Yes)."
      },
      "skills": [
        "Symmetry of Covariance under Swapping Coordinates"
      ],
      "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": "協方差必須綁定同一完整有限joint law。操作=covariance-symmetry；資料={\"support\":[[-2,-1],[-1,0],[0,1],[1,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]]}。請依 [Cov_XY,Cov_YX,equal] 欄序回答。",
        "en": "Use the complete list of distinct integer pairs=[[-2, -1], [-1, 0], [0, 1], [1, 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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "joint law total mass := sum(1/8, 1/4, 1/8, 1/2)",
      "after": "joint law total mass = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 joint law total mass，不以小數近似替代。",
          "en": "Recompute joint law total mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/8",
          "1/4",
          "1/8",
          "1/2"
        ],
        "output": "1",
        "label": "joint law total mass"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "original distinct joint pairs=[[-2, -1], [-1, 0], [0, 1], [1, 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",
      "relation": "implies",
      "rule_id": "qualify_whole_joint_law_preserve_alignment",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只用同一共同機率空間的完整原始聯合表，保留順序、配對與零質量；每個質量非負且精確總和1才可求協方差。",
        "en": "Use the complete original joint table on one common probability space,preserving order,pairing and zero masses.Covariance requires nonnegative masses with exact total1."
      },
      "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 atom0 weighted X := mul(1/8, -2)",
      "after": "original atom0 weighted X = -1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom0 weighted X，不以小數近似替代。",
          "en": "Recompute original atom0 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/8",
          "-2"
        ],
        "output": "-1/4",
        "label": "original atom0 weighted X"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "original atom0 weighted Y := mul(1/8, -1)",
      "after": "original atom0 weighted Y = -1/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom0 weighted Y，不以小數近似替代。",
          "en": "Recompute original atom0 weighted Y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/8",
          "-1"
        ],
        "output": "-1/8",
        "label": "original atom0 weighted Y"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "original atom0 XY := mul(-2, -1)",
      "after": "original atom0 XY = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom0 XY，不以小數近似替代。",
          "en": "Recompute original atom0 XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "-1"
        ],
        "output": "2",
        "label": "original atom0 XY"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "original atom0 weighted XY := mul(1/8, 2)",
      "after": "original atom0 weighted XY = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom0 weighted XY，不以小數近似替代。",
          "en": "Recompute original atom0 weighted XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/8",
          "2"
        ],
        "output": "1/4",
        "label": "original atom0 weighted XY"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "original atom1 weighted X := mul(1/4, -1)",
      "after": "original atom1 weighted X = -1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom1 weighted X，不以小數近似替代。",
          "en": "Recompute original atom1 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "-1"
        ],
        "output": "-1/4",
        "label": "original atom1 weighted X"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "original atom1 weighted Y := mul(1/4, 0)",
      "after": "original atom1 weighted Y = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom1 weighted Y，不以小數近似替代。",
          "en": "Recompute original atom1 weighted Y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "0"
        ],
        "output": "0",
        "label": "original atom1 weighted Y"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "original atom1 XY := mul(-1, 0)",
      "after": "original atom1 XY = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom1 XY，不以小數近似替代。",
          "en": "Recompute original atom1 XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "0"
        ],
        "output": "0",
        "label": "original atom1 XY"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "original atom1 weighted XY := mul(1/4, 0)",
      "after": "original atom1 weighted XY = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom1 weighted XY，不以小數近似替代。",
          "en": "Recompute original atom1 weighted XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "0"
        ],
        "output": "0",
        "label": "original atom1 weighted XY"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "original atom2 weighted X := mul(1/8, 0)",
      "after": "original atom2 weighted X = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom2 weighted X，不以小數近似替代。",
          "en": "Recompute original atom2 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/8",
          "0"
        ],
        "output": "0",
        "label": "original atom2 weighted X"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "original atom2 weighted Y := mul(1/8, 1)",
      "after": "original atom2 weighted Y = 1/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom2 weighted Y，不以小數近似替代。",
          "en": "Recompute original atom2 weighted Y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/8",
          "1"
        ],
        "output": "1/8",
        "label": "original atom2 weighted Y"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "original atom2 XY := mul(0, 1)",
      "after": "original atom2 XY = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom2 XY，不以小數近似替代。",
          "en": "Recompute original atom2 XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "original atom2 XY"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "original atom2 weighted XY := mul(1/8, 0)",
      "after": "original atom2 weighted XY = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom2 weighted XY，不以小數近似替代。",
          "en": "Recompute original atom2 weighted XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/8",
          "0"
        ],
        "output": "0",
        "label": "original atom2 weighted XY"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "original atom3 weighted X := mul(1/2, 1)",
      "after": "original atom3 weighted X = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom3 weighted X，不以小數近似替代。",
          "en": "Recompute original atom3 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "1"
        ],
        "output": "1/2",
        "label": "original atom3 weighted X"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "original atom3 weighted Y := mul(1/2, 2)",
      "after": "original atom3 weighted Y = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom3 weighted Y，不以小數近似替代。",
          "en": "Recompute original atom3 weighted Y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "2"
        ],
        "output": "1",
        "label": "original atom3 weighted Y"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "original atom3 XY := mul(1, 2)",
      "after": "original atom3 XY = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom3 XY，不以小數近似替代。",
          "en": "Recompute original atom3 XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "original atom3 XY"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "original atom3 weighted XY := mul(1/2, 2)",
      "after": "original atom3 weighted XY = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original atom3 weighted XY，不以小數近似替代。",
          "en": "Recompute original atom3 weighted XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "2"
        ],
        "output": "1",
        "label": "original atom3 weighted XY"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "original E_X := sum(-1/4, -1/4, 0, 1/2)",
      "after": "original E_X = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original E_X，不以小數近似替代。",
          "en": "Recompute original E_X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1/4",
          "-1/4",
          "0",
          "1/2"
        ],
        "output": "0",
        "label": "original E_X"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "original E_Y := sum(-1/8, 0, 1/8, 1)",
      "after": "original E_Y = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original E_Y，不以小數近似替代。",
          "en": "Recompute original E_Y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1/8",
          "0",
          "1/8",
          "1"
        ],
        "output": "1",
        "label": "original E_Y"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "original E_XY := sum(1/4, 0, 0, 1)",
      "after": "original E_XY = 5/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original E_XY，不以小數近似替代。",
          "en": "Recompute original E_XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4",
          "0",
          "0",
          "1"
        ],
        "output": "5/4",
        "label": "original E_XY"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "original product of means := mul(0, 1)",
      "after": "original product of means = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original product of means，不以小數近似替代。",
          "en": "Recompute original product of means with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "original product of means"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "original covariance := sub(5/4, 0)",
      "after": "original covariance = 5/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original covariance，不以小數近似替代。",
          "en": "Recompute original covariance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5/4",
          "0"
        ],
        "output": "5/4",
        "label": "original covariance"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "original:E_X=0;E_Y=1;E_XY=5/4",
      "after": "Cov=E_XY-E_X*E_Y=5/4",
      "relation": "implies",
      "rule_id": "covariance_same_joint_three_moments",
      "conditions": [],
      "explanation": {
        "zh-Hant": "三個期望全部使用相同原子與質量，不能混合不同邊際模型，亦不假設獨立。",
        "en": "All three expectations use the same atoms and masses.Do not mix marginal models or assume 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": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "swapped atom0 weighted X := mul(1/8, -1)",
      "after": "swapped atom0 weighted X = -1/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom0 weighted X，不以小數近似替代。",
          "en": "Recompute swapped atom0 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/8",
          "-1"
        ],
        "output": "-1/8",
        "label": "swapped atom0 weighted X"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "swapped atom0 weighted Y := mul(1/8, -2)",
      "after": "swapped atom0 weighted Y = -1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom0 weighted Y，不以小數近似替代。",
          "en": "Recompute swapped atom0 weighted Y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/8",
          "-2"
        ],
        "output": "-1/4",
        "label": "swapped atom0 weighted Y"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "swapped atom0 XY := mul(-1, -2)",
      "after": "swapped atom0 XY = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom0 XY，不以小數近似替代。",
          "en": "Recompute swapped atom0 XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-2"
        ],
        "output": "2",
        "label": "swapped atom0 XY"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "swapped atom0 weighted XY := mul(1/8, 2)",
      "after": "swapped atom0 weighted XY = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom0 weighted XY，不以小數近似替代。",
          "en": "Recompute swapped atom0 weighted XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/8",
          "2"
        ],
        "output": "1/4",
        "label": "swapped atom0 weighted XY"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "swapped atom1 weighted X := mul(1/4, 0)",
      "after": "swapped atom1 weighted X = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom1 weighted X，不以小數近似替代。",
          "en": "Recompute swapped atom1 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "0"
        ],
        "output": "0",
        "label": "swapped atom1 weighted X"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "swapped atom1 weighted Y := mul(1/4, -1)",
      "after": "swapped atom1 weighted Y = -1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom1 weighted Y，不以小數近似替代。",
          "en": "Recompute swapped atom1 weighted Y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "-1"
        ],
        "output": "-1/4",
        "label": "swapped atom1 weighted Y"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "swapped atom1 XY := mul(0, -1)",
      "after": "swapped atom1 XY = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom1 XY，不以小數近似替代。",
          "en": "Recompute swapped atom1 XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "0",
        "label": "swapped atom1 XY"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "swapped atom1 weighted XY := mul(1/4, 0)",
      "after": "swapped atom1 weighted XY = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom1 weighted XY，不以小數近似替代。",
          "en": "Recompute swapped atom1 weighted XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "0"
        ],
        "output": "0",
        "label": "swapped atom1 weighted XY"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "swapped atom2 weighted X := mul(1/8, 1)",
      "after": "swapped atom2 weighted X = 1/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom2 weighted X，不以小數近似替代。",
          "en": "Recompute swapped atom2 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/8",
          "1"
        ],
        "output": "1/8",
        "label": "swapped atom2 weighted X"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "swapped atom2 weighted Y := mul(1/8, 0)",
      "after": "swapped atom2 weighted Y = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom2 weighted Y，不以小數近似替代。",
          "en": "Recompute swapped atom2 weighted Y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/8",
          "0"
        ],
        "output": "0",
        "label": "swapped atom2 weighted Y"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "swapped atom2 XY := mul(1, 0)",
      "after": "swapped atom2 XY = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom2 XY，不以小數近似替代。",
          "en": "Recompute swapped atom2 XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "swapped atom2 XY"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "swapped atom2 weighted XY := mul(1/8, 0)",
      "after": "swapped atom2 weighted XY = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom2 weighted XY，不以小數近似替代。",
          "en": "Recompute swapped atom2 weighted XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/8",
          "0"
        ],
        "output": "0",
        "label": "swapped atom2 weighted XY"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "swapped atom3 weighted X := mul(1/2, 2)",
      "after": "swapped atom3 weighted X = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom3 weighted X，不以小數近似替代。",
          "en": "Recompute swapped atom3 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "2"
        ],
        "output": "1",
        "label": "swapped atom3 weighted X"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "swapped atom3 weighted Y := mul(1/2, 1)",
      "after": "swapped atom3 weighted Y = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom3 weighted Y，不以小數近似替代。",
          "en": "Recompute swapped atom3 weighted Y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "1"
        ],
        "output": "1/2",
        "label": "swapped atom3 weighted Y"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "swapped atom3 XY := mul(2, 1)",
      "after": "swapped atom3 XY = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom3 XY，不以小數近似替代。",
          "en": "Recompute swapped atom3 XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "swapped atom3 XY"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "swapped atom3 weighted XY := mul(1/2, 2)",
      "after": "swapped atom3 weighted XY = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped atom3 weighted XY，不以小數近似替代。",
          "en": "Recompute swapped atom3 weighted XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "2"
        ],
        "output": "1",
        "label": "swapped atom3 weighted XY"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "swapped E_X := sum(-1/8, 0, 1/8, 1)",
      "after": "swapped E_X = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped E_X，不以小數近似替代。",
          "en": "Recompute swapped E_X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1/8",
          "0",
          "1/8",
          "1"
        ],
        "output": "1",
        "label": "swapped E_X"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "swapped E_Y := sum(-1/4, -1/4, 0, 1/2)",
      "after": "swapped E_Y = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped E_Y，不以小數近似替代。",
          "en": "Recompute swapped E_Y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1/4",
          "-1/4",
          "0",
          "1/2"
        ],
        "output": "0",
        "label": "swapped E_Y"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "swapped E_XY := sum(1/4, 0, 0, 1)",
      "after": "swapped E_XY = 5/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped E_XY，不以小數近似替代。",
          "en": "Recompute swapped E_XY with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4",
          "0",
          "0",
          "1"
        ],
        "output": "5/4",
        "label": "swapped E_XY"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "swapped product of means := mul(1, 0)",
      "after": "swapped product of means = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped product of means，不以小數近似替代。",
          "en": "Recompute swapped product of means with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "swapped product of means"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "swapped covariance := sub(5/4, 0)",
      "after": "swapped covariance = 5/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 swapped covariance，不以小數近似替代。",
          "en": "Recompute swapped covariance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5/4",
          "0"
        ],
        "output": "5/4",
        "label": "swapped covariance"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "swapped:E_X=1;E_Y=0;E_XY=5/4",
      "after": "Cov=E_XY-E_X*E_Y=5/4",
      "relation": "implies",
      "rule_id": "covariance_same_joint_three_moments",
      "conditions": [],
      "explanation": {
        "zh-Hant": "三個期望全部使用相同原子與質量，不能混合不同邊際模型，亦不假設獨立。",
        "en": "All three expectations use the same atoms and masses.Do not mix marginal models or assume 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": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "original covariance=5/4;swapped covariance=5/4",
      "after": "equal by XY=YX and E_X*E_Y=E_Y*E_X",
      "relation": "implies",
      "rule_id": "covariance_swap_both_roles_symmetry",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交換兩變數時同時交換均值角色，乘法交換律讓乘積期望與均值乘積都不變。",
        "en": "Swap both variable and mean roles.Commutativity leaves both the product expectation and product of means unchanged."
      },
      "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": [
      "5/4",
      "5/4",
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(5/4, 5/4, 是)",
    "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-r05-covariance-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261026-r05-covariance-foundations/questions.jsonl",
      "batch_sha256": "bbd1b50d44aff10183fe3178034a74eede1636496e7e219652bc8417573a0746",
      "original_line": 36,
      "original_line_sha256": "4f75643523589dcd57ea69641883665e2bdb2600cd52eb451714e432c1945ef1",
      "question_compact_sha256": "15ffc3b428f006efc6917be6660becf0a03ce04c79ea4d570a671e35ea1de510",
      "english_row_compact_sha256": "9db06db5a137481d4b4b9e58c1c5c926104661dc2dd1efeca76675a5dd3155c6",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261026-r05-covariance-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261026-r05-covariance-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-covariance": {
        "sha256": "3b0480b894efd088fcbcb2c9c856fb6e46bc8b62a223285723b060bf0070143d",
        "english_sha256": "df82b21fa123d4f51a7320f03eab21717099356a4a1b8959e5cec7b86c26e42a",
        "original_path": "public/data/mko/covariance.json",
        "english_path": "public/data/i18n/editions/mko-covariance.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "c6ae588be9bc8b3dd8461be32c073abfa25a0f827b4602cf28bf291ce532d507",
    "original_candidate_sha256": "4fe35fc38610b09c18722dbe19d89ce0f429e01dedfafdc76510cea93f155369",
    "original_candidate_line_sha256": "4fe35fc38610b09c18722dbe19d89ce0f429e01dedfafdc76510cea93f155369"
  },
  "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-p24",
    "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": "f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff",
      "question_first_witness_sha256": "49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 47,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "b334cc892c553d2d8adba6a003b9991845bc3c076b71c3017d40d0d69bdfb7d0",
      "arithmetic_steps": 43
    },
    "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."
  }
}